Research Laboratory of the Portland Cement Association Bulletin 11 Shrinkage Stresses in Concrete Part 1— Shrinkage (or Swelling), Its Effect upon Displacements and Stresses in Slabs and Beams of Homogeneous, Iso- tropic, Elastic Material Part 2 — Application of the Theory Presented in Part 1 to Experi- mental Results By GERALD PICKETT March, 1946 ( *m< AGO Authorized Reprint from Copyrighted Journal op the American' Concrete Institute New Center Building, 7400 Second Boulevard Detroit 2, Michigan Jm. and Feb. 1946, Proceedings Vol. 42, pp. 165-204 and 361-400 Title 48-8 — o port of PROCEEDINGS, AMERICAN CONCRETE INSTITUTE Vol. 42 JOURNAL AMERICAN CONCRETE INSTITUTE (copyrighted) Vol. 17 No. 3 7400 SECOND BOULEVARD, DETROIT 2, MICHIGAN January 1946 Shrinkage Stresses in Concrete* By GERALD PICKETTt Member American Concrete Institute SYNOPSIS Theoretical ex] :ttions of < and slabs that occur during the course of drying and expressions for distribution of the accompanying shrinkage stresses are derived in Part 1. These expressions an on the assumption that the laws governing the development of shrinkage stresses in concrete during drying are analogous to those governing .-.pment of thermal stresses in an ideal body during cooling. Three cases are considered: (a) slab or beam drying from one face <>! (b) slab or beam drying from two oppo and (c) prism drying from four fat The applicability of the equations to concrete is considered in Part 2 (to appear ACI Journ It is shown that the course of shortening of prisms is in very good agreement with the theoretical equations and that from a test on one prism the shortening versus period of diving of other prisms of the same mat. rial differing in size and number of sides exposed to drying can be predicted with fair accuracy if the differences in size are not too great. However, it is shown thai the theory must be modified to take into account inelastic deforma- tion and to permit the supposed constants of the mat .rial to vary with moisture content and size of sp if the theory is to be in agree- ment with all results on all types of specimen of a given concrete. Various tests are described which, when used in conjunction with the theory, provide a means foi g some of the more fundamental properties of concrete and for predicting the performance of concrete under some conditions in the field INTRODUCTION Concrete, like many other materials, gains or lost- nth changes in ambient conditions. With each change in water con 1 concrete ♦Received by the Institute. April 30, 1945. 1 Professor ot Applied Mechanic* Kansas State < oUege, Manhattan, kans.; formerly Portland Cement Association Research Laboratory, Chicago. 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SHRINKAGE STRESSES IN CONCRETE 167 the distribution of shrinkage stresses in, concrete beams and slabs during the course of drying. Second, to show by means of data from specimens under controlled conditions the manner and degree to which the equations apply to con- crete. Third, to suggest methods for studying some of the more fundamental properties of drying concrete. No attempt will be made here to give a complete analysis of stresses in concrete. In particular, the effect of aggregate particles on the stresses within the hardened paste will not be considered. Before expressions for shrinkage stresses in concrete can be derived, assumptions must be made in regard to the relation between shrinkage and moisture content and the laws controlling the flow of moisture in concrete as well as the relation between stress and strain. The actual relationships are not as simple as could be desired. If the flow of water were entirely by vapor diffusion, if the vapor pressure of the water in the concrete were proportional to the moisture-content, and if permeability were independent of the moisture-content, then the differ- ential equation for the flow of water would be a partial-differential equation known in physics and mathematics either as the diffusion equation or as the equation of heat conduction. Carlson, 1 * in a study of distribution of moisture in concrete, assumed that this equation applies. If the flow of water could be expressed by the diffusion equation and if the shrinkage (or swelling) tendency! of each elemental volume were linearly related to the moisture-content, the unrestrained shrinkage (or swelling) could also be expressed by the diffusion equation. This possi- bility was also considered by Carlson. 1 But the flow of water is different from that indicated by the diffusion equation, and the relationship between the change in moisture-content and unrestrained shrinkage is not linear as required by these equations. Moreover, satisfactory expressions for either the flow of water or the moisture-shrinkage relation have not been found. It is believed that moisture in concrete flows partly as liquid in capillar- ies, partly as vapor, and partly as adsorbed liquid on the surface of the colloidal products of hydration. While drying progresses, the vapor pressure of the water remaining in the region losing water decreases progressively with the moisture content. This change in vapor pressure with change in moisture content is not linear with respect to moisture *See references at end of text of Part 1. fBy shrinkage (or swelling) tendency is meant the unit linear deformation due to any cause other than stress that would occur in an infinitesimal element if the element were unrestrained by neighboring elements. It is not to be confused with the average unit deformation, commonly called shrinkage, of a so-called un- restrained specimen, nor with the resultant linear unit deformation which for the x-direction will be de- signated ?z. Hereinafter, the linear unit shrinkage tendency will be referred to either as unrestrained shrinkage, for clarity, or merely as shrinkage, for brevity. JOURNAL OF THE AMERICAN CONCRETE INSTITUTE January 1946 . ther i- the rate <«f How proportional to the gradient id \ and relative proportions of the sp upied by liquid : a drying proceeds. Thia fact, ae weX\ na the non-uniformit a believed to be partly responsible for the bich vapor pressure depends on moisture content and the way in win- ■■ gradient of vapor pressure. ght-loea relation is different for different ritioo of the concrete and the conditions , Bferenl during first shrinkage from subsequent volume changes, [fasatui the ratio <>i change of length to lo At first, comparatively Bmall cha jiit . The higher the water-cement I be smaller the change during ' < ■ omes imi<-li larger and i i w bich ii may cither increase j,i [t is believed that ■ bat bold 1 1n* \\ai. I e Bhape, cles; and (c) th< 'It oi the • red in any at udy concrete !«■' • «i.M'> i rom t In hrough t he si • ^e> ImiI h ;.prc»\ima1i T\ l:i\\ tO In- ;i — Mined, hilt m-, ill h< and foi 1 \1< .nd \\hil< • SHRINKAGE STRESSES IN CONCRETE 169 the diffusion equation applies to shrinkage even though the simple rela- tions that are implied by that assumption are contrary to fact. It is fur- ther assumed that concrete follows Hooke's law. The derivations given in Part 1 are based upon these assumptions. Since in Part 1 the derivations for deformations and stresses are based on the assumptions that shrinkage follows the diffusion equation and the material follows Hooke's law, the equations are even more applicable to thermal stresses in metals than to shrinkage stresses in concrete. In fact, much of the mathematical work given here was taken from the literature on diffusion of heat and on thermal stresses, as the references will show. However, certain corresponding coefficients in the two problems are of an entirely different order of magnitude. For example, the numerical value of the thermal diffusivity for steel expressed in square inches per second is approximately the same as the numerical value of the shrinkage diffu- sivity of concrete expressed in square inches per day. Because of the relatively slow diffusion of shrinkage the application of the hypothesis to the shrinkage of concrete necessitates the study of early transient condi- tions (usually ignored in the treatment of heat). PART 1— SHRINKING (OR SWELLING), ITS EFFECT UPON DISPLACEMENTS AND STRESSES IN SLABS AND BEAMS OF HOMOGENEOUS, ISOTROPIC, ELASTIC MATERIAL Notation S = free, unrestrained unit linear shrinkage-strain ~S = free, unrestrained unit linear swelling-strain 5 ro = final shrinkage-strain under fixed ambient conditions, value of S when t = a> S av = average shrinkage over the volume of the specimen, the same as average shorten- ing per unit length if the material follows Hooke's law t = time in days k = diffusivity coefficient of shrinkage in sq. in. per day / = surface factor, characteristic of the material and the boundary conditions, in in. per day a, 6, c, d, I = distances related to the dimensions of the specimen in inches B = fb/k, a non-dimensional parameter T = kt/b 2 , a non-dimensional parameter B c and T c , non-dimensional parameters corresponding to B and T and used when a second characteristic dimension of the specimen must be considered x, y, z = rectangular coordinates n = nth root of ptan = B $ m = same as /3 n except used in connection with c, whereas n is used in connection with b A „ = Fourier coefficient 2B „ 2£, F n = , F m =~ & + B + j3» Bl + B c + fil b -f„, a. -A $1 el H n = — F m ff m = — F„ 170 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE January 1946 V»A 2 J <r t = normal components of stress parallel tox-, y-, and z -axes— positive if tensile, negative if compressive. = elongations inx— , y— , and 2 -directions Txv, *». T r , = shearing-stress components y*,, 7*z, Ty» = shearing-strain components E = Young's modulus in u = Poisson's ratio r = deflection in inches, displacement of the elastic line in the y-direetion -N = the normal to the surface directed outward rX -x= *(*) = 2 rl P(x) = — = j e fa t the probability integral 2 r 00 _ X 2 - -m fa = 1 — - 1 cos n - cc -7^ COS 0* - *< = 1 - - F m - 1 cos &» H b 0= -r« 1 - - #. 1 H. = 1- Equation for diffusion of unrestrained shrinkage The diffusion equation is a mathematical statement of the fact that for each infinitesimal volume - the excess of the substance in question flowing in over that flowing out per unit of time is equal to the rate of mcr ; ft <* in that volume. When similar assumptions are k in regard to shrinkage, shrinkage thus being treated as if it were a ;ust as heat is so treated, the result is- '' La* 1 ty 1 az' J " where fc is the diffusivity of shrinkage. The equation becomes ..(1) k 5 -**—« 2 SHRINKAGE STRESSES IN CONCRETE 171 at exposed boundaries and ^= (3) dX at sealed boundaries wherr N is the normal to the surface, directed away from the body / is the surface factor S oo is the value that S will eventually reach under fixed ambient conditions. Equations 2 and 3 correspond, respectively, to Newton's law of cooling at exposed boundaries and to no flow of heal 3 erfectly insulated boundaries in the analogous problem of flow of heat. If the boundaries of the body are not parallel planes, a transformation of Equation 1 from an expression in rectangular coordinates to some other form is usually desirable. For i . if the bod si circular cylinder, Equation 1 is best transformed to Ldr* r dr r 2 d9 2 d z- J dt where r, 6, and z are cylindrical coordinates. Frequently, the condition — = at some boundaries or some other conditions will make S inde- dX pendent of certain coordinates and thereby simplify Equation 1. Since the form of the solution for S depends upon the form of the differential equation, the form of the solution is dependent upon the boundary condition and the shape of the body under investigation. The initial conditions (values of S at t = 0) and any variation in boundary conditions with time will also affect the form of the solution. Assumptions as to elastic properties After a satisfactory solution for S has been obtained, then displace- ments and stresses will be found by the application of certain funda- mentals of the theory of elasticity. The solutions for stresses are here restricted to homogeneous isotropic solids that follow Booke'a law. Also, as will be brought out below, the effect of Poisson's ratio will be neglected in some cas Effect of shape of body on relative values of principal stresses The state of stress at any point in a body is defined by the directions and magnitudes of the three principal stresses. The three principal stresses in wide -labs and in narrow brain- will be in the directions of length, width, and depth, respectively, if the bodies are under uniform exposure either from one or from two opposite face- and are without external restraint. The principal stress in the direction of depth (normal • 1946 tresses will be D - MBMfled t<» «i«*pt It the prinri- I -« VM.ltl « - W ill Im M ON z SHRINKAGE STRFSSES IN CONCRETE 173 Equation 2a is also satisfied if fi n is I he nth root of (3 tan 13= f - (2b) n i.e., j8 B tan Pn = ~ (2c) A The above statements may be verified by substituting S from Equation 4 into Equations la, 2a and 3a. For time t = » , Equation 4 reduces to S = S co , which is in accord with the definition of 5 oo . An infinite series of terms such as the trigonometric series in Equation 4 is necessary to give an arbitrary distribution of shrinkage at time t = 0. If the initial conditions are such that S = when t = 0, then the Fourier coefficients A n are given by* ^n = 2 / " 6 O co tC S n (fbV + Jg , U/ k cos$ n (fb\ It therefore follows that V COS^ s sr -t£ „ h Si where = 1- Te'S.-/ (5) £— cos p n F = 2B ' n 5 2 + B + ft Equation 5 (in slightly different form) and similar equations for other shapes and other conditions, applied to analogous phenomena, may be found in the literature of mathematical physics such as the textbooks of Byerly, Carslaw, and Ingersoll and Zobel. Various tables and diagrams have been prepared from which the numerical relationship of the four non-dimensional quantities S/S «, y/b, B, and T may be found, such as Fig. 4, page 841 of Perry's Chemical Engineer's Handbook (1934), * T he general procedure of obtaining Fourier coefficients to satisfy initial conditions semewhat analogous tO tin- present problem ifl given in Articles 66 to 68 of Byerly'a Fourier Series and Spherical Harmonics on: Ginn A Co., 1893). 174 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE January 1946 To use more than a few of the terms in Equation 5 for the evaluation of S/S » is very laborious because of the difficulty in evaluating ft, and F n . The number of terms required for a given degree of precision will depend somewhat upon the parameters B and y/b but is chiefly controlled by the parameter T. Computations show that very little error is introduced by neglecting all terms in the series except the first if T is more than about 0.2; but many terms are needed for the usually desired precision if T is less than 0.02, — the smaller the value of T the greater the number of terms needed. Very precise values of S/S oo for small values of T may be found without the tedious computation indicated in Equation 5 by using another expression which will now be derived. Solution in terms of the probability integral As long as the shrinkage at the sealed surface remains negligible, the distribution of shrinkage from the exposed surface inward will be nearly independent of the distance between the two surfaces. Suppose that instead of considering the surface at y = to be sealed, the body is considered to be extended to infinity in a negative w-direction. Then instead of the boundarv condition — =0 at y dy = 0, the requirement will be S = (6) at y = — oo . The solution* satisfying Equations la, 2a and 6 and giving S = when t = Ois S_ Sc = [1-Jj [>-?] _2Vf . - _ 2 V T _ B(l |) + B*~T ■ {') where <j> (x) is — = / e dx and T is again used in place of — . The quantity 1 — o ■ ■■ -4 f dx, is known as the proba- bility integral. Values of <t> [z) may be readily found by using a table of the probability integral. Numerical calculations show that Equation 7 gives values that differ from those given by Equation 5 by an amount less than the value of S/*Sa: at y =0; therefore, Equation 7 may be used in place of Equation 5 whenever T i> so small that S Sm at y = is less than the permissible error. *Thi*> solution it very simiLar to that given for an analogous problem bv Carslaw in Article 25 of The Maciiiillan <k Co.. Ltd., 2d e<l., 1921). SHRINKAGE STRESSES IN CONCRETE 175 Table 2 and Fig, 7, showing S/S « in terms of y/b, kt/b 2 , smdfb/k, were prepared from Equations 5 and 7. Stresses and strains Continuity, Hooke's law and equilibrium. As stated previously, the solutions for stresses are here restricted to homogeneous, isotropic solids that follow Hooke's law. Equations for the stresses that would be pro- duced in such a body by the shrinkages given by Equations 5 or 7 will now be derived. The shrinkage S has been denned as the linear unit deformation that would occur if each infinitesimal element were unrestrained. However, the properties of a continuous solid will not permit an arbitrary distribu- tion of deformations ; therefore, unless the distribution of shrinkage given by Equation 5 happens to be compatible with the conditions of continuity, stresses will be produced that will modify the deformations so as to make them compatible. Although in general six partial differential equations are required for a complete mathematical statement of the conditions of compatibility, 3 these are reduced to **- (8) dy 2 for either long narrow beams (plane stress) or slabs (plane strain) if the stresses are considered to be independent of the longitudinal coordinate x. The term e x is defined as the resultant unit deformation in the z-diree- tion (the direction of length). It is therefore the algebraic sum of shrink- age, S, and the strain produced by stresses. a v is obviously zero; and if Poisson's ratio is zero or if the discussion is confined to narrow beams, a t is negligible. Therefore, e x = &-S (9) E or, solving for stress, <r, - E(e t + S) (10) where E is Young's modulus, The restriction imposed by Equation 8 requiring that the expression for longitudinal deformation contain no terms in y other than the first power (second derivative equal to zero) is equivalent to the assumption usually made in the elementary theory of beams that "plane cross-sections remain plane." If longitudinal restraint is complete, then e x is zero and it follows from Equation 10 that <r x = ES. If, however, longitudinal short- ening is permitted but complete restraint against bending is provided, then e x is not zero but is still independent of y. If the beam has no external restraint, the non-symmetrical distribution of shrinkage causes it to warp, making e x a linear function of //. For no external restraint the equations of 176 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE January 1946 equilibrium (summation of forces in the j-direction equal to zero and summation of moments about the z-axis equal to zero) become / <j z dy = 0. (11) and / <*xV dy = (12) It may be shown by substituting Equation 10 into both Equations 11 and 12 that if shrinkage (S) is either independent of or a linear function of y an unrestrained beam will be free of stress (e I = - S and c x = 0). For any other variation of shrinkage a stressed condition must result because the restriction on e x (Equation 8) will not permit it to be equal and oppo- site to S if shrinkage is a non-linear function of y. The only solution for e x that satisfies Equations 8, 10, 1 1 and 12 is b b = ( 6 v- 4 )i/ 5 ^ + ( 6 - 12 i)i /■**■<*> When this value of e x is substituted into Equation 10, the result is = E o o (14) Finally, 5 from Equation 5 may be substituted into Equation 14 thus g Tf, J 1 " 6 ! 8 /, 11 * n u *"-" ;'»""> »>eam as a function of the parameters V, o, kt/b,fb/k, 5 c and of Young's modulus. This substitution will not be made until later, because it seems advisable at this time to consider another approach. Solution by sup, rposition. Although the above derivation is short and is in the simplest form for checking the mathematical correctness of the equation, a derivation in which elementary solutions are superposed is also desirable because it will be easier in general to understand and because the hnal _ expresMon, are in more usable forms. In this second derivation the resultant stress ... U cowddered as consisting of three parts. The first part s that stress which would be produced by complete restraint against longrtudmal deformation; the second part is a uniform stress equal to and opposite m s,gn to the average of the first part; the third part is a stress SHRINKAGE STRESSES IN CONCRETE 177 resulting from a simple moment that is equal to and opposite in sign to the moment produced b\ r the sum of the first two parts. That is, the first part alone <j x f would result from complete restraint, the sum of the first and second parts <r/ would result from restraint against warping only; the sum of all three parts, i.e., a XJ would result if no external restraint were applied during shrinkage. Although in this derivation an expression for <r x appears to be the ultimate goal, expressions for a J and for a/ are also desirable. The stress trj may be representative of the stress in pavement slabs or building walls that are restrained from shortening and the stress <r/ is representa- tive of an unrestrained wall drying equally from two opposite sides (Case II discussed later). Since for complete longitudinal restraint e x = 0, it follows from Equa- tion 10 that the first part of the stress is aj = ES (15) i r Since the average value of trj is — I aj dy, the second part of b e r a x is — r- I S dy; therefore, the sum of the first and second parts (<r/) is given by a x " = E -\f Sdy • (16) The moment produced by <r/ is the moment necessary to prevent warp- ing. This moment per unit width of beam is found by multiplying Equa- tion 16 by y dy and integrating. This gives i- b M j *; ydy = E J Sy dy - -J I Sdy L .(17) For no external restraint this moment must be removed by superposing an equal and opposite moment. The stress resulting from a moment — M is given by the elementary theory of beams as M {y - 6/2) 1/12 ¥ JOL. CONCRETE INSTITUTE January 1946 T 1 I / / L -I .•s N t!nT i ■ , ., I V / Ir Ofllhff'l - ■ teaM SHRINKAGE STRESSES IN CONCRETE 179 in terms of the deflection iw it produces, the expressions for the stresses are put into more usable forms. When this is done, the following equations for stresses are obtained. For complete longitudinal restraint (first part of <r x ), a x > = ES (15) For restraint against warping only (sum of first and second parts of Oi <r x " = E (S - S av ) (20) For no external restraint (sum of all three parts of <r»), / y\ 26^ • (21) Evaluation of the parameters ~ and — ^ When Equation 5 for S/Sc is substituted in Equation 18 for shortening and in Equation 19 for warping and the indicated integrations are performed, the result is b °° 1 .(22) 2b Vrri 3l*S< i r s s av >y- = bj sZ ydy " 25. = 2- -rft e G. (23) where and H n = 2B 2 G, 01 (B*- + B + fil) \cosB n 2 / 81 If T t the non-dimensional time-factor, is small, the series in Equations 22 and 23 converge rather slowly, and in that case it is convenient to use the following equations obtained by substituting Equation 7 into Equa- tions 18 and 19, respectively.* Sa e i <t>(B V T) - 1 + =- .(24) The lower integration limit for each integral is decreased from to — »; 180 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE January 1946 2bv„ WS< JL I_ 2B + B* v* r>...(25) Furthermore, if the parameter B V T is very small, it is still better to use the following equations obtained by expanding the expressions in the brackets of Equations 24 and 25. ^~ = BT 4 _ 1 _ ■= B V T +~(B V TY 2bv n 3Vt V BT 15 V 8 ,— (BVr) 3 + 3Z 2 S, (24a) flr / b\/ 4 _ 4 ^^i^v *■>•)- . (25a) In general the following rules will be found applicable for rapid evalua- tion of the parameters |= and ^t to a fair degree of accuracy. 0<x> Ot^O m If T is more than about 0.05, use Equations 22 and 23. If T is less than about 0.05 and B is more than about 5, use Equations 24 and 25. If T is less than about 0.05 and B is less than about 5, use Equations 24a and 25a. Forces and moments necessary for complete restraint. The force neces- sary for longitudinal restraint is JV dA. Therefore, the average force b per unit area is j-J,,' dy. From Equations 15, 5, and 22 this becomes force per unit area = ES t 2 e II. ■ (26) From Equations 17, 5, and 23, the moment per unit width necessary tor restraint against warping is found to be °° -Tfi U = £Sc & ^ e G n 1 (27) SHRINKAGE STRESSES IN CONCRETE 181 Simplification by taking B as equal to infinity. The principal equa- tions derived above reduce to simpler forms and the computation of numerical values is less tedious if the assumption is made that B, i.e., fb/k } equals infinity. If B is large, say 100 or more, the error introduced by assuming it to be infinity is negligible. However, if B is less than about 5, the error introduced by considering it to be infinity may be appreciable as is shown, for example, by Fig. 8, 9, and 14. Whether justifiable or not, the assumption that B = oo is frequently made in anal- ogous problems to which the diffusion equation applies. This assump- tion was made by Terzaghi and Frohlich 4 in developing the theory of settlement of foundations due to consolidation of underlying material, by Glover 5 in a study of distribution of temperature in concrete dams, and by Carlson 1 in a study of distribution of moisture and shrinkage in con- crete. The more important of the above equations for the special case of B = cd are given below : Equation 5 becomes S ^-4(-l)- 1 -(2n-l)*jT cos (2n - 1) — . s (2fl - l)ir 2b' Equation 7 becomes Equation 22 becomes Sa Sc __8 s- 1_ " 7T 2 Z. (2n - - (2h - I) 2 (7r 2 /4) T I) 2 Equation 24 becomes v -J T O oo \ 7T Equation 23 becomes 2bv m WSc = z 7T 2 (2n - \y i Equation 25 becomes 2bv ma: 3l 2 Sc yr . w (2n - 1) - 1 _ (2n - I) 2 — T 4 T . 182 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE January 1946 Tables, curves, and computations.* Tables and diagrams such as those by Newman 1 are available from which values of N S« and S m /Sd may be determined. However, such published tables arc in general not ade- quate for the present problem. The smallest value of the parameter T used by Newman in his computations was 0.1, whereas the stresses in cone sired for a much earlier period. The tables given here were prepared for T as low as 0.001. Moreover, so far as is known, er — ; not previously been evaluated fortius or any analogous problem and its bion is necessary for the problem here ■ oi Equations 15, 20, and 21 for the theoreti- one side under the different modes of bh i luce quantities ■ , ' ; and - • ion ol ih« three parameters y b, B and I ig 7 Tin' second quan- Ild V i- given in Table 'A and shown ni:i!)til \ - '" B£ a function of B and • >/ ■ . s M\ in Fig 9. Tables . r > and 6 20 and 21 as functions of i be prepared after the three 1 abl< - 2, •'». and 1 1 Results foi 0, li. L2, and 13 Fig. l i shows rsufi the parameb on <»i the tables and diagrams iaf ions based upon the i c used instead ol 1 lie mtered. I mple, q 7 then l ) ( u)i 1.1/ 9.1 / • • »'•' ' " ■■•'■<■> ' .. .-.:... -...:.'.. | ,„••.! c i' » *-r :>!,!.► *.»♦- :,|, J COliM ;.U:\ SHRINKAGE STRESSES IN CONCRETE 183 From mathematical tables i op; 4> (1) = 0.15730; 4> (1.5) = 0.03389; e = 3.4903 Therefore S/Sa> = 0.1573 - 0.03389 X 3.4903 = 0.0390. Note that this is the value given in Table 2 for the above values of B } T, and y/b. Also note that for the same B and T the table gives zero for y/b = 0, showing that it was permissible to use Equation 7 instead of Equation 5. When the theoretically correct equations are used, the computations are more involved. For instance, let T = 0.1 instead of 0.01 in the above example. T will then be so large that S/Sa> will have an appreci- able value at y/b = 0. Therefore, Equation 7 will not be applicable and Equation 5, the exact equation, must be used. A substitution of values for T and y/b into Equation 5 gives 00 —QlQl A = i _ V F cos °- 8 ^ So* 2i ( " co$p n 1 The first step in evaluating the above expression is to determine n which Equation 2c shows to be a function of fb/k and n, i.e., B } and the integer n. The determination of |3 n by interpolation is simplified by the introduction of a„ where a n depends on B and n. The equation for 0„ is then written n = (n - 1 +a n )7r (28) Curves of a n versus B for the first six values of n and for n = 21 are shown in Fig. 2. By means of Fig. 2 and Equation 28 any desired j3 n may be found with reasonable accuracy for any value of B, The first six values of n for several different values of B are given in Table 1. After finding n for the given values of B and n, the factors F nt cosp n , cos (fin-) and e ~ T &* are determined. F n and cos /3 n as func- tions of B are shown in Fig. 3 and 4, respectively, for the first six values of n. The functions cos i & n -\ and e ~ l $ n are readily obtained from mathematical tables after the products 0« ~ and T &l have been deter- b mined. When the proper values of the four factors listed above are sub- S stituted, the above equation for^— becomes O as 184 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE January 1946 0.84147 X 0.3152 X 0.4966 _ 0.1965 X 0.2161 X 0.9963 O oo 0.2541 0.00844 X 0.1286 X 0.7280 0.8101 0.6277 This reduces to — = 1 - 0.5183 - 0.0674 - 0.0009 = 0.4134 The values for S/S » given in Table 2 were computed by one or the other of the methods illustrated above In preparing Table 3 from_which Fig. 8 was constructed (Fig. 8 shows shortening as a function of V T for various values of B), values of H n in Equation 22 were needed. Values of H n as a function of B for the first six values of n are shown in Fig. 5. In like manner, Fig. 6 showing G n as a function of B, served in the preparation of Table 4 from which Fig 9 was constructed. Of course, for small values of T, Fig. 5 and 6 are not necessary since either Equation 24 or Equation 24a is used instead of Equation 22 and either Equation 25 or Equation 25a is used instead of Equation 23, depending on the value of B. Application to beams or slabs of any width-to-depth ratio when Poisson's ratio ts not zero. The effect of Poisson's ratio was neglected in the pre- ceding derivations. Its effect stated in general terms in the introductory remarks in regard to Case I will now be analvzed in more detail. If Poisson s ratio is not zero, Equation 9 for e, and Equation 10 for c x will be modified to include the effect of c t . That is, '*-w. = E(e. + S). (10a) However, if the ratio of width to depth is small, „ z will be negligible and ■ tion 10a reduces to Equation 10. On the other hand, if the ratio of width to depth is very ] ar ge (a slab), the width being comparable with tlK-Je^th, then a. will be equal to „,. If „, = ,„ then Equation 10a °' = T~- (e * + V (10b) t J'Tn? 1 -' difference u between E^on 10b for a wide slab and Equa- .on 10 or a narrow beam is the factor * which occurs in Equation 10b bul ,,„ . Equation 10. Therefore, for stresses in a slab, E k ! Equations 15, 20, and 21 is replaced by -£.. The stresses in beams „ dth- to-d..p,h nmo i, in„.r„K,lia, ( . will have stresses intermediate between of narrow beams and of slabs. Since ,o, appear in Equa- SHRINKAGE STRESSES IN CONCRETE 185 tion 22 for average shrinkage nor in Equation 23 for warping, these quantities are the same for narrow beams and wide slabs. CASE II— SLAB OR BEAM DRYING FROM TWO OPPOSITE SURFACES Equations taken from those derived for Case I. Since the flow of mois- ture in a slab drying from only one surface is believed to be the same as that in either half of a slab of twice the depth drying from two op- posite surfaces, it will be assumed that the theoretical equations de- rived for shrinkage of a beam or slab drying from only one surface will apply equally well for either half of a beam or slab drying from two opposite surfaces. The plane midway between the drying surfaces will be taken as the plane y = as shown in Fig. 1 for Case II. Since the two halves of the beam will mutually restrain each other from warping, the equations for stresses, strains, and shortening in each half will be the same as those given previously in Case I for a beam restrained against warping and drying from one surface. CASE HI— RECTANGULAR PRISM DRYING FROM FOUR FACES Shrinkage The differential equation and boundary conditions. For a prism drying from four faces but not from the ends the diffusion equation reduces to \dy 2 dz 2 / dt The exposed faces of the prism will be taken as the planes y = =*= b, z = ± c, as shown in Fig. 1 for Case III. The boundary conditions then become ^ = ±L(Sa> - S) (2a) dy k at the boundaries y = =*= b and 3? = *£(fl. - S) (3b) dz k at the boundaries z = =*= c. The solution satisfying Equations lb, 2a, 3b, and giving S = at t = and S = S*> at t - °o is the following: -m cos A^ir" - r «* cosfij- O oo I COS f} n 1 Ye F m c cos p„, . (5a) 1 where p m F m} and T c correspond to p n , F n , and T, respectively, the differ- ence being that the dimension b has been replaced by c. 186 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE January 1946 Shrinkage expressed in terms of the solutions given for a prism drying from one face or two opposite faces. Since the infinite series in the first bracket in Equation 5a is identical with the one given in Equation 5 and the infinite series in the second bracket is like the first except that y is replaced by z, b by c, etc., and since Equation 5 applies to either half of a slab exposed on two opposite surfaces (Case II), it follows that the brackets have the following values: '. - - = 1 - & , 2e ■T e & cos f} n cos m - c cos fi m (29) = 1 - * .(30) where </> b is the value S/Sa> would have if only the surfaces y = ± b were exposed and <f> c is the value S/Sa> would have if only the surfaces z = =±= c were exposed. A substitution of Equations 29 and 30 into Equation 5a gives 5— — 4>b + 4>c O 00 <t>t>4>c- .(5b) Equation 5b shows that the evaluation of shrinkage for a prism dry- ing from four surfaces becomes a problem of adding the independent effects of drying from surfaces that are perpendicular to each other and then subtracting a term proportional to the product of the separate effects. For example, consider the shrinkage tendency at the point y = 0.46, 2 = 0.8c in a prism for which c = 26 (width equal to twice the thickness)' Let/, k, and t be such that/6/fc = 5.0 and kt/b* = 0.20; then fc/k = 10.0 and kt/c* = 0.05. </> 6 is found in Table 2 or from Fig. 7 to be 2398 Since Table 2 was prepared for B equal to 0.1, 1.0, 5.0, and » only, and since fc/k = 10, 0, can be found from Table 2 only by interpolation However, examination of Table 2 indicates that for kl/c* equal to 05 Equation 7 can be used instead of Equation 5 without appreciable error and therefore the equation rather than the table will be used to obtain <i> c - From Equation 7 = | ' r | ~ * (■ °^— + 10 V0.05^ V2V0.05/ \2V0.05 / 2+5 .2V0.05/ V2V0.05 / From tables giving probability integrals and the exponential function 4>c = 0.5273 - 0.000156 X 1097 = 0.355 Therefore at y = 0.46, z = 0.8c, and t = 0.20 b-,'k s — = 0.2398 + 0.355 - 0.2398 X 0.355 = 0.510. SHRINKAGE STRESSES IN CONCRETE 187 Shortening expressed in terms of the shortening of a prism drying from one face or from two opposite faces. The average shrinkage S at is given by b c s av = ~tff Sdy " : (18a) From Equations 5a and 18a -r# ro -T e 0L 5fL_ = i - 5" e H n V e H m (22a) 1 1 or ^- « H h + H c - HJI* (22b) O CO where H b is the value S S would have if only the surface y = ± b were exposed and // ifi the value S ar /'Sao would have if only the surfaces z = ± c were exposed*. Therefore, the average shrinkage, and conse- quently the shortening, if the bod ic, of a prism drying from four sides may be found by considering the separate effects of drying from opposite -ides in pa; For example, consider the shortening of the prism discussed above. H h is found in Table 3 to be 0.3510 and H c is found to be 0.1753. Therefore, — = 0.3510 + 0.1753 - 0.3510 X 0.1753 = 0.4- 7 O QD Stresses and strains Nature of the problem and the method to be used to obtain a solution. In -■■s I and II previously discussed, where shrinkage was a function of time t and only one space coordinate y and where the problem was further simplified by neglecting the effects of the length and the width of the imen on the distribution of stresses, a solution for the one stress involved was readily obtained. However, in the problem now under con- sideration, a prism doing from four surfaces, shrinkage varies with an additional coordinate z. As a result stresses vary with this additional coordinate also, and more than one stress will be involved. The problem will be somewhat simplified by neglecting the effect of the length on the distribution of st e., the assumption will be made that stresses do not var>- along the length. The distribution of given by the solution based on this assumption will deviate a negligible amount from •If a prism were drying from all six surfaces, the corresponding equation would be SWS. = H. + Hh — He - IUH- - //-// f - H-H* — HaHiH,. Another way of expressing these relations is given by Glover VRdLI 188 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE January 1946 the theoretically correct distribution when stresses in the central portion of a long prism are under consideration (principle of Saint Venant 8 ). The solution for shrinkage in terms of two space coordinates was air. when only one coordinate was involved (Equation 5a for I II compared with Equation 5 for Cases I and II j, because tendency to shrink is considered to be a scalar quantity. On the other hand, since stresses are tensor quantities, the solution for stresses usually becomes much more complicated whenever more than one coordinate is involved. In fact, elastieians have obtained exact solutions meeting all boundary conditions for only a relatively few problems in which stresses were func- tions of at least two coordinates and then only by considering the body to be infinite in the direction of one of these coordinates. The difficulty is that since stresses are tensors, boundary forces are vectors, and in two-dimensional problems two components of force must be satisfied at each boundary. The specified conditions of stress at anv two opposite boundaries can be satisfied by superposing particular solutions of the differential equations in accordance with the usual methods of Fourier analysis. But, in general, solution* ing rigorously the boundary requirements at two pairs of opposite boundaries simultaneously cannot be found by the usual metho A method of solving such problems after the appropriate differential equations have been derived was explained by the author in a recent paper*. That method will be used here. It is about the same as that u*ed previously by Taylor" and by Timoshenko" in analogous problems. Der ' the differential equations rehr , shrinkage Bv neglecting the variation oi and strains along the length of a body the problem becomes a two-dimensional problem in plane strain The flowing equations taken from the theory of elasticity are then applica- Equations of Equilibrium: <>y dz + a^ =0 Condition of Compatibility: Modified to Include Isotropic Shrinkage: dz 2 ' ~ E k ~ ^ 9 " ^ - S " - s - in ;hi ^ problem _ SHRINKAGE STRESSES IN CONCRETE 189 € v = — [— v.<*x + <r v - M°"z] " - 5 e g = — [ — /x^r — mc v + ^] - E - s 2 (1 + M ) 7 U * = T M E The above seven equations giving relations between the eight unknown stresses and strains can be reduced to the following two equations by eliminating the four strains and the shear stress: <r M = M (<r ¥ + O +E(S- S av ) ■ (31) y »(,, + a: ) = JL_ V *S (32) 1 - n where V 2 is written for — H . By* dy 2 These two equations together with the two equations of equilibrium and the boundary conditions that fri and r yz = at y = ±6 V «" 1,J - '&z cr z and T yz = at z = =*= c and Equation 5a for S constitute the mathematical statement of the problem. Solution for stresses. In general the stress a x will be larger than either ff v or ov The stress r vz will be relatively small in all cases. If only the value of the theoretical maximum stress is desired, a fairly good approxi- mation can be obtained by the following simplified formula: <r x = E (S - S av ) approx (31a) where S is given by Equation 5a and S av is given by Equation 22a. If, however, an accurate theoretical value of all stresses is desired, a complete solution must be obtained and this is given below: The solution for <r yt <r zy and r yz meeting all of the above requirements is as follows:* r 2 _ q 2 2 b ^- *r~ An *»= A °-lzt+~c Z Z -f^sa iy cosy ]Z t=i i=i ♦Equations 33. 34. and 35 for stresses and Equations 36 and 37 for the coefficients B, and Ci given here are alnWt the same as Equations i, 2, and 3 for stresses and Equations 6 and 7 for the coefficients B n and A n MW*y4?&ento&L%™ The Aperies and the Antenna enter into the equations, given here in place of the S™?n the stress S given there; otherwise, except for slight differences in notation, the correspond, in^ Pauftfon^ are identical and the equations given here may be derived by the procedures given there. Equation 38 for X? and Equation 39 for A may be verified by substituting Equations 33 34, and 5a ». t Equa^on32andth e np^^ resulting equation within the domain under consideration by the usual Fourier analysis. 190 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE January 1946 , \ r, cos yjZ r • . / 2- B >' T^Th ™ y SUlh ™ ~ ( 1+ y* b coth y& cosh y 3 y] i=i cosh y } b Z r cos dig r cosh ^ a%Z + ^~ aiC €oth a < c ) cosh Oiz] . . (33) , & 2 -3^ 00 oo ,_i, l^ + i Z Z 7T C0S ^ cos 7 ; z »*=1 3 = 1 Z„ COS 7,2 r . , . '' c^sT^i biy smh ™ + (1 - T ' 6 co ^ ^ 6 ) cos * r,y] i=i \ r cos a f y r . , Z 'cosh af [aiZ Slnh aiZ ~ C 1 + a ' c co ^ «.<) «wA a, 2 ] (34) t'=l oo oo > — r sin any sin y,z 00 4- 'S" R Sin y > 2 I l , Z *' cos* 7i 6 ™ 6 c<rfA ^' 6 s '^ *» - 7,2/ cosA 7/ y] i=i 00 Z Ci cosA a,c LttiC C0 ^ a ' f s ''' i/! a ' z ~ «* cosh afi] . 1=1 ■ (35) where a,- = iw b ' T/-2 _ _c_ AoJ-iy , 6 6 7T F+}I«t-z<->^ te«A l -= B,= i'=l i= 1 i^io i + *(< m c, c ) (36) SHRINKAGE STRESSES IN CONCRETE 191 c A.(-iy c (-iy ~ - y (_i) 1" £■ — c, = i=i tan/i- — i=i i + (S)' B< 6 V h^fcott™ to llll (37) .1., = " = 1 '" = 1 v"*«") v "ft"/ n = 1 m = 1 The above equations, together with Equation 31 for a SJ constitute a complete solution on the basis of the given assumptions. In general, if both the parameters T and T c are equal to or greater than 0.1, the series given above converge very rapidly so that only a few terms need be taken for a good approximation. The example given below will demonstrate the use of the above equations. Example: Stress at the middle of one side of a square prism for which B equals 5 and at a time for which T equals 0.1. If only one term in each series is used, the following values are obtained: From Equations 38 and 39 and Table 1. ES " (0t + 0i)e -O-lfl.-O- 1 * // A = A. 1-M ES c (1.3138* + 1.3138*) 0.8415* X 0.9130* .4. = - 2.0377 i<:s 1- ES<» (1.3138* + 1.3138*) 0.8415 2 X 0.9130 2 An — ~ — — " — : — / _? \2 (1 - *0 ** (l - _£-)' V 1.3138V 192 A n = - 0.0093 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE January 1946 E a oo l-M When these values are substituted into Equation 36, the result is (_ 2 _^ +0.0093) ^-'«5*£ Cl B l = 1 + 7T (cotf* 7r — /an/i 7r) or since Ci = B Y (square prism), C, = B,= ~ 0.1472 — When the above values and z = c, y = Q are substituted into Equation 33, the result is + 0.0093 - 0.1472 X 0.3583 + 0.1472 X 0.97731 ^* Jl- M = p.0377 L 6 2 =C y = ■[ + 0.3396 + 0.0093 - 0.0527 -r 0.1439 - ■ = 0.4401 « z = c 2/ = 1-M When the summation of each series is carried to two terms, the result is ES 0,4214 - — — and when the summation is carried to four terms in each 1- ES : series the result is 0.4221 t^Jfl f or this stress. 1- M The above shows that the series converge very rapidly for this example. The stresses a t and r vz are obviously zero at the point under considera- tion. From Equations 31, 5b, and 22b, Tables 2 and 3, and the above result for c y , = [j-j- 0.4221 + (0.0221 + 0.6913 - 0.0221 X 0.6913) - (0.2186 + 0.2186 - 0.2186 X 0.2186)1 £S«> z = c y = Li - M 0.4221 + 0.3087 ES ] 2 = C y = o For all values of Poisson's ratio M less than 0.212, the above expressions will give larger values for <r y than for *„ This fact is of interest because, 5E STRESSES IN CONCRETE 193 in |bi - - OTHER METHOOS D tO - ■ I M SUMMA- I ng, warp- I r beam & na one ^ ntaina an | n the correspond!!* 194 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE January 1946 equation for a slab or beam drying from two opposite surfaces. Com- putations show that the equation with the added term gives much less stress tor the same size of body and the same period of drying. Compare / Tables 6 and 5. " ' Equations are given for all the stresses in a prism drying from all four surfaces, the third condition treated. These equations, though rather complicated in appearance, can be readily evaluated if the desired accur- acy is such that only a few terms in each series need be considered. For a rather rough approximation of the stress that is usuallv the most impor- tant in this third condition, the comparatively simple equation <r, = E(S ~ Scv) is recommended. Tables and curves are given from which the theoretical shrinkages, stresses, etc, may be obtained, at any point in the specimen after any period of drying, for various values of the physical properties, diffusivity surface factor, ultimate shrinkage, and dimensions of the specimen. Examples are given showing how numerical values mav be computed from the equations and how the tables and curves may be used. REFERENCES JaLFe?. m g 7 S &\ ?e ^,^T27 COI1Crete MeB ^ n '" »* * "■ Carlson, ACI Jolr.val, ^^^'^^^^^^trst^hk and E - s - Sokolnikoff ' 3. Theory of Elasticity, by S. Timoshenko, McGraw-Hill. New York, 1934, p. 196 Fran, S^^S^^St^" * * * *«*» "* °. K ' ™** <rfCtenri^a?"'^ T ^ e GIo^ D ArT i T U,i0n in a v SucC ^ ion 0{Ulis Due t0 ReI ^ « "TkrT * „ OIo%er ' ACIJovR * A L. ^v.-Dec. 1937,Proc.. V.34, p 105 In£. 0f T C h heS O Enl., i^ P g Tl9 S 31 lidS ' , ' b> ' * * Xwman ' Interim Publication, Am! "Flaw of Heat in DanV' by R. E. Glover, ACI Journal. Xov,Dee. 1934, Proc. 8. Reference (3), p. 31. pp 222^" ° fPtaUS ^ SMh - ** S - Timoshenko, McGraw-Hill, Xew York, 19M,' 12. Ref. (3), pp. 21-23. C^^bfati^p t JX%^^S^S!' T^T-? 1 fi*— » Concrete 14 "A «JLi ArTi j , 1 ^ r ° c ^ m - • ,Jf Testing Matenals, V. 39, 1939, p. 913. ss?" "*£& ass ra/raaarafstat SHRINKAGE STRESSES IN CONCRETE 195 17. Relaxation Methods in Engineering Science, by R. V. Southwell, Oxford Univ. Press, 1940. 18. "A Lattice Analogy for the Solution of Stress Problems," by Douglas McHenry, J. Inst, of Civil Engrs., Dec. 1943, pp. 59-82. 19. "Numerical and Graphical Method of Solving Two- Dimensional Stress Prob- lems," by H. Poritsky, H. D. Snively, and C. R. Wylie, Jr., J. Applied Mechanics, June 1939, V. 6, No. 2, p. A-63. 20 "The Effect of Change in Moisture-Content on the Creep of Concrete under a Sustained Load," by G. Pickett, ACI Journal, Feb. 1942, Proc. V. 38, p. 333. 21. "Die Beeinflussung des Schwindens von Portlandzement durch Sulfate," by G. Haegermann, Zement 28 (40) 599 (1939). 22. "A New Aspect of Creep in Concrete and Its Application to Design," by Douglas McHenry, Proc. Am. Soc. Testing Materials, V. 43, 1943, p. 1069. 23. "Attempts to Measure the Cracking Tendency of Concrete," by R. W. Carlson, ACI Journal, June, 1940, Proc. V. 36, p. 533. 24. "The Influence of Gypsum on the Hydration and Properties of Portland Cement Pastes," by W. Lerch, unpublished. 25 "The Dependence of the Shrinkage of Portland Cement on Physical and Chem- ical Influences," by H. Kuhl and D. H. Lu, Tonind. Z. 59 (70) 843 (1935). 196 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE January 1946 m • 7T~" 3_ Fig, 1 — Illustrations of the con- ditions treated. Shading indi- cates sealed surfaces. Case I — Beam (or slab) drying from one face only. Case II — Beam (or slab) drying from two op- posite faces. Case III — Prism drying from four faces. The ends of the prism at x = * a are sealed. OSOp 0.45 - 040 - OiS - 050 - n 0« - 020 - o ig - 010 - 005 - / 0-0 »here (J n is the n?> root ^^ * P n ton(i n .fjL^ B i «y / y // X / / v%/ ^/ S SS; * — -^^^ = ss: ^--^^ B (loq let Fig. 2 — Curves for the determination of /3„ Fig. 3— Relationship between F, and H SHRINKAGE STRESSES IN CONCRETE 197 B (log scale) Fig. 4 — Relationship between cos 0„ and B B (log scale) Fig. 5 — Relationship between H n and B ato - 0.09 - 0.08 - ao? - o.ot> - n 0.05 - 0.04 - 0.03 - o.oz B ( log scale Fig. 6 — Relationship between G n and B 198 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE January 1946 T T~ f- A- T IT \ $ . P > I" |\ \~ \\_ ' |\K\ \ ^^^$\ V- l^^^^^sA 1 S? S 2 S g $ g S 2S° t * C fc. Bj 0) o JI *. a "J ^; o-£ 2*o a> o " o a> o Is o o- SI 8 _ 0> o.S .¥ c Oi ^ o o OJ -C t] I - tf J-V. ., 1 1 c ;" 1 5 ■is* lit i \ 5^<- 1 m j/y / j i jf f I ' 1 ^7 \ \ d 4 --_ d o 5 - o « °"£ « o o >• *-6 I* -C D m M O — E o o o aj *s*ie 1 ' \\ ,\ o \ \ \ II DO \ \ J 1 5 i i 1 1 Vj ■ o CC3 ci \ o o q k ^ - CC ^ >s. ^ > \ k \ X ^ X A \\ \v c \ V \ ^\ in QQ s \ 1 o o C$1 >/ .. h -a c o CQ *o o o c O -0 SHRINKAGE STRESSES IN CONCRETE 199 X <* £| 8 <S^ 1 8 ^-? II it* c 1 1\ t / J \ \ V oVo\\ Y K X V uO o || l-s fc i o o z a o a o o o hi O o> c a- » " E 0) o (M 1 1 1 1 $ 8 c3 «s ^ -J + _o - 4k' Ol 3^K <j i t f 1 — o ^"V ■\ f \ / c j \0 V o/ <y o o * II ■*« .2 o t o.E O >s 6 E = .2 «l «!! I E *0 3. T H a* -§-? ^i ri- $ c L^. — #-/ r \\\ Z7Z^ C -SjLi ^M-< i sj^ _t< i 1 O 2C O-D •z a> .>• ^ ' = s "5 5 0) — I ° §i 8 4^ - ^ :» _-<j . 6 <0^ -if - ^ - -4? r 6 .! 5 J l! j Q \ S o »i K b k. **^&^ 5 i - k. c . ■ o 5 > D L 8 "5 it 3 O "5 200 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE January 1946 ||5* . Fig. 14-Maximum stress (maximum value of stress at exposed surface) and maximum warping vs. the parameter B TABLE 1-0„ n tan n = B B 0.0 0.1 0.5 1.0 2.0 5.0 10.0 100.0 0.311053 0.653271 0.860334 1.076874 1.31384 1.42886 1.55525 7T 2 3.1731 3.2923 3.4250 3.6436 4.0338 4.3058 4.6656 3tt 2 ft 2tt 6.2991 6.3615 6.4372 6.5783 6.9097 7.2281 7.7760 5tt/2 0% 3tt 4, 5w 9.4354 12.574 15.715 9.4774 12.606 15.740 9.5292 12.645 15.771 9.6296 12.722 15.834 9.8927 12.935 16.010 10.200 13.213 16.260 10.887 13.998 17.109 7tt 2 9*72 llir/2 _^ N CONCRETE 201 TABLE 2— RATIO OF SHRINKAGE (OR SWELLING) TO ULTIMATE SHRINKAGE (OR SWELLING) / ' I • ' I JOURNAL OF THE AMERICAN CONCRETE INSTITUTE January 1946 TABLE 3-UNIT SHORTENING-AVERAGE UNIT SHRINKAGE (|= ) 1 v ~ T ^u = 1 - 2 e H„ 0.1 B = 0.5 B = 1.0 2.0 5.0 B = 10.0 B = 0.001 0.002 0.003 0.005 0.010 0.015 0.02 0.03 0.04 0.05 0.075 0.10 0.15 0.20 0.30 0.4(1 0.50 0.75 1.0 1.5 2.0 3.0 4.0 5.0 7.5 10.0 15.0 20.0 O IIOIIl 0.0002 0.0003 0.0005 O OIIK) 0.0015 0.0020 0.0030 0.0039 0.0049 0.0074 0.0098 0.0146 0.0193 0.0288 0.0382 0.0474 . 0702 0.0924 0.1353 0.1761 0.2521 0.3211 0.3837 O .-.161 0.6301 0.7658 B556 0.0005 0.0010 0.0015 0.0024 0.0048 0.0072 00U5 0.0141 0.0185 0.0231 0.0340 O 0446 0.0653 0.0854 0.1239 0.1606 n 1057 0.2771 0.3502 0.4751 0.5767 0.7233 0.8182 0.8821 0.0594 0.9860 i) 9983 i» gggg O 0010 0.0019 0.0029 0.0047 o 009:1 0.0137 0.0181 0.0265 0.0347 0.0426 0.0620 0.0803 0.1154 0.1489 0.2064 . 2666 0.3188 0.4339 0.5296 0.6751 o 7756 0.8930 (I 04 SM . 9756 0062 . 9994 1 oooo 0.0019 0.0037 0.0055 0.0090 0.0173 0.0252 0.0317 0.0473 0.0611 0.0742 . 1050 (I 1336 . 1860 0.2327 0.3190 0.3939 4604 II 5062 O 0970 . 8306 . 9052 O 0625 O 0907 0.9971 0.9998 1 . 0000 0.0045 0.0085 0.0124 0.0196 0.0370 0.0515 . 0649 0.0891 0.1115 0.1319 0.1779 0.2186 (i 2895 0.3510 0.4555 0.5422 0.6148 0.7498 0.8375 0.9315 0.9711 . 9949 0091 it 9996 1.0000 0.0080 0.0147 0.0206 0.0323 0.0556 . 0755 . 0932 0.1242 0.1512 0.1753 . 2282 0.2739 0.3510 0.4167 0.5258 0.6136 . 6849 0.8109 0.8865 0.9591 0.9857 0.9981 0.9998 1.0000 0.0357 0506 0.0619 0.0800 0.1129 . 1383 0.1596 0.1954 0.2257 0.2523 0.3090 3506 . 4370 504 1 0.6133 0.6979 0.7641 0.8726 0.9313 0.9800 0.9942 0.9995 1.0000 TABLE 4- WARPING (%~ in thousandths) 0.0010 0.0015 0.0020 0.0030 0040 0.0050 0.0075 0.010 0.015 n 020 0.030 (HO 0.050 0.075 0.10 0.15 0.20 0.30 0.40 o 50 0.75 1.0 1.5 2.0 3.0 4.0 5.0 7.5 10.0 15.0 20.0 B = 0.1 (I 04S 0.071 0.093 0.137 0.180 0.222 0.324 0.421 0.606 0.778 1.092 1.375 1.628 2.161 2.568 3.126 3.457 3.757 3,843 3.851 3 782 3.694 3.519 3.353 3.044 2.763 2 50 s 1.969 1 506 0.953 26 IW 2 1 -T$\ B = 0.5 B = 1.0 2.0 B = 5.0 0.235 0.348 (I 15s -.74 883 1 (Ills 1 579 2 04 5 2.920 3.735 5.415 6.518 7 s:i2 10.138 1 1 . S46 14.017 15.128 15 773 15 546 15.043 L3 588 12 217 o 869 7 059 5.203 3.417 2.216 0.763 0.262 0.031 004 0.46 0.68 0.90 1.32 1.72 2.12 3.06 3.94 7.08 9.74 12 05 14.03 17.96 20 75 23.96 25 29 i:* 61 24.14 22.59 is S5 15.67 10.82 7 17 3 . 57 1.70 0.81 . 1 3 02 0.00 0.91 1.33 1.73 2.54 3.31 4.03 5.74 7.35 10.25 12 92 17.34 21.07 24.21 30 os 33.87 37.56 38.32 36.28 32.88 29 43 22.07 16.52 9.25 5 is 2.05 0.51 0.16 0.01 0.00 2.13 3 os 3.99 5.73 7.23 8.69 13 Os 1 5 . 1 s 20 4 5 24.97 32.63 38.32 42.70 49.98 53.81 55.70 53.89 47.03 39.90 33.64 21.86 14.20 5.09 2.53 0.45 0.08 0.01 0.00 3.82 5.43 S.89 9.61 12.03 14.27 19.16 23.35 30.38 35.92 44.71 50.93 55.33 62.22 65.16 64.50 60.56 50.52 41 .38 33 77 20.27 12.17 4.38 1.58 0.20 0.03 0.00 16.83 19.79 23 . 22 27.90 31.66 34.89 41.36 46 . 42 54.12 67.67 72.80 76.15 79.76 80.40 74.18 66.85 52.74 41.27 32.25 17.40 9.39 2.74 0.80 0.07 0.01 0.00 , SHRINKAGE STRESSES IN CONCRETE 203 TABLE 5— RATIO OF STRESS a" T IN AN UNRESTRAINED BEAM DRYING FROM TWO OPPOSITE SIDES (OR IN A BEAM DRYING FROM ONLY ONE SIDE AND RESTRAINED AGAINST WARPING) TO THE ULTIMATE STRESS FOR COMPLETE RESTRAINT ES*> S Sa Si v y y y V y y V V y v V T -=0 -=0.2 -=0.4 -=0.6 -=0.8 - = 1.0 -=0 -=0.2 -=0.4 -=0.6 -=0.8 -=1.0 b b b b 6 6 & b 6 b 6 b B=0.1 B = 1.0 0.0 0.005 -.0005 -.0005 -.0005 -.0005 -.0003 -.0047 -.0047 -.0047 -.0040 .0703 0.010 -.0010 -.0010 -.0010 -.0010 -.(1001 -.0093 -.0093 -.0093 -.0091 .0000 .0942 -.0O15 -OKI 5 -.0014 ,0006 .0120 -.0137 -.0137 -.0125 .1101 0.020 -.0020 -.0020 -.0020 -.0017 .0013 .0138 -.0181 -.0148 1 24 .1235 0.030 -.0030 -.(11)20 0019 2 -.0265 -.020 1 -.0170 .0246 .1425 0.(11(1 -.0030 -.0030 -.0019 .0041 .01 s.3 -.0346 -.0344 ;| 1 -.0165 ,1563 0.060 -.0049 -.00 is -.0019 .0051 .0198 -.0123 -nil', -.0361 -.0152 <<l I 1 .1670 0.075 -.0072 -.0000 -.001 » .0072 .0228 -.0570 -.0112 -.0104 0616 .1851 0.10 -.0090 -.OOSl -.1.110 1 .Hi .on -.0682 -.0054 .1961 0.15 -.0117 -.0106 -.0069 .oooo .011 1 .0277 -..010 -.0821 .001 1 .0871 .2064 0.20 -.0133 - 0119 -.0074 .0005 .0293 -oooo -.0880 -.0539 .0929 .2081 0.30 -.0149 -.0132 -.0079 .0010 .013S .0306 -.1028 -.0900 -.0530 ,0089 .0948 .2027 0.40 -.0154 -.0136 -.OOS] .0011 .0141 ,0308 -.0855 -.0497 01)0 1 .1893 i >..-,() -.0154 -.0080 0013 .0142 .0311 -.001 1 -.0S01 .0088 .1766 0.75 -.0152 -.0133 -.0079 .0013 .0302 -.0764 -.0669 .0074 .0699 .1471 1.0 -.0148 -.0130 -.0(i77 .0012 .0295 -.0634 -.0555 -.0321 .0062 .0582 .1222 1.5 -.0140 -.0123 -.0072 .0011 .0131 ,0281 -.0439 -.0384 -.0223 .0401 .0844 2.0 -.0134 -.0118 -.0070 .0011 .0124 .0267 -.0266 -.0153 .0029 .0277 .0583 3.0 -.0 122 -.0107 -.0003 .0010 .0113 .0243 -.0145 -.0127 -.0074 0013 .0131 .0277 4.0 -.0111 -.0007 -.0057 .0102 .0221 -.0069 -.0060 -.0035 ,0007 .0133 5.0 -.0101 -.on.v.t -.0052 9 0092 .0200 -.0032 -.0028 -.0016 .0001 .0030 (.004 7.5 -.0070 -.0069 -noil 0006 .0073 .0157 ion -.006] -.0055 -.0033 .0055 .0120 15.0 -.0038 -.0033 -.0020 .0035 ,0076 20.0 -.0023 -.0021 -.0012 .0002 .0022 .0047 B = 5.0 £ = «> 0.0 0.001 -.00 1". -.00 15 -.0045 -.0045 .1514 -.1.357 -.0357 -.0357 -.0357 .9743 0.002 -.OOS5 -.0OS5 -.0085 -.0085 -.0084 .2010 -.0506 -.0506 .9494 0.0(15 -.0100 -.0196 -.0100 -.0124 .2823 -.0800 -.0800 -.0345 .9200 0.010 -.0370 -.0370 -.0370 .0020 -.1129 -. 1 1 20 - L127 .0412 .8871 0.015 -.0515 -.0515 -.1383 -.1383 -.1378 -.1171 .1100 .8617 0.020 -.0649 -.0646 -.0523 .4120 -.1596 -.1595 -.1141 .1577 3404 0.030 -.0890 -.Os.sl -.Osls -.0538 .0871 .4441 -.1944 .8046 0.040 -.1 100 -.1098 -.0998 -.0481 .117., .1000 -.2249 -.2210 -,10 1s -.0684 .2538 .7713 0.050 -.12! tO -.1277 -.1001 - 0401 .1404 .4707 -.2492 -.2408 -.1945 -.0464 .2748 .7477 0.075 -.17(10 -.1604 -.1207 -.0200 .1765 .4778 -.2892 -.2683 -.0073 0.10 - 1965 -.1700 -. L230 -.0044 .1948 .4727 -.3096 -.1771 .0151 it. 15 - 2160 -.1969 -.llso -.0130 .2052 t hi -.3012 .0317 1 ». _'( 1 -.2164 -.ISO 7 -.1112 -.0199 -.2765 -.2404 .4959 0.30 -.lols -.H.71 -.0948 .0216 .17 10 .3551 -.2200 -.1000 -.1048 .0292 .1986 .3867 0.40 -.1634 - 1 1J-' -.OSOI .OI01 .1485 -.1723 -.1492 -.0818 .0231 ,3021 0.50 -.1370 -.1 103 -.01,7 1 0103 -.1347 -.1166 -.0640 .0180 .2359 (J. 7.'. - osoo -.0770 -.0437 0106 -.0727 -.0629 .0655 1.0 -.0582 -.0506 -.Ojsj .1064 -.0393 -.0341 -.01S7 .0353 .0687 -.0246 -.0214 - 0170 .0223 .0448 -.0114 -.0100 .0015 .0200 2^0 -.0104 -.0051 . (HII2 .0094 -.0034 -.0029 -.0016 .0004 .0030 .0058 -.0019 -.0016 -.0009 .0016 204 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE January 1946 TABLE 6— RATIO OF STRESS <r, IN AN UNRESTRAINED BEAM DRYING FPniu ONLY ONE SIDE TO ULTIMATE STRESS FOR COMPLETE RESTRAINT g? 0.0 0.005 0.010 0.015 0.10 0.30 1.5 0.001 - -f= + («- 12*) Ox s 2bv. v v y -=0.2 -=0.4 -=0.6 b b b v y = 1.0 6 6 B = 0. 1 y „ y y y y y -=0 -=0.2 -=0.4 -=0.6 -=0.8 - = 1.0 b b b b b b .0021 .0071 .0003 .0005 9 .0011 .0011 .0003 .0003 .0001 .0001 .0001 .0000 - -.0019 28 -.0017 -.0014 -.0014 -.0005 -.0016 -.0015 - .0001 .0003 .0001 .0001 78 .0100 n - - .0060 .0030 - B 1.0 0080 .0198 .0343 .o;uo .0419 .0510 .0473 0441 .0210 .0145 .0029 .0049 .0074 .0087 .VO\'0 .0089 .0042 .0030 .0022 .0014 - .0010 .0001 -.0022 -.0046 -.0070 -.0091 -.0135 -.0169 -.0193 -.0226 -.0237 -.0243 22 - . ' -.0102 -.0160 -.0133 - - -.0031 -.0015 -.0006 -.0072 -.0138 -.0192 -.0233 -.0310 -.0320 -.0320 -.0303 -.0274 -.024 7 -.0218 -.0199 -.0183 -.0132 -.0120 -.0061 -.0030 -.0013 -.0006 -.0116 -.0142 -.0142 -.0131 -.0106 -.0081 -.0061 -.0031 -.0012 .0008 .0019 .0026 .0025 .0020 .001b .0011 .0002 .0002 .0001 .0576 .0706 .0766 .0811 .0641 .0840 .0773 .0716 .0626 .0490 .0445 .0318 .0340 .0282 .0195 .0135 .0063 .0031 .0015 B = 5.0 .0141 -- .0011 - .0001 .0000 - 5 - -.0004 - - 28$ 3 27 7 .0000 " .1142 72 .0006 B = 887 - 728 >8S .0014 .0411 .0105 .0027 .0001 -.0009 -.0006 -.0002 .0002 .0000 .0000 -.1044 -.1031 -.0641 -.0416 -.0021 ,0006 - 781 -.121b -.1626 -.1*23 -.1741 78 -.1031 -.0814 57 -.0459 -.0340 -.0111 -.0018 -.0006 -.0963 -.1326 -.1601 -.00b3 .0005 .0093 .0117 .0119 .0112 .006S» .0053 .0029 .0015 .0004 .0001 .8101 .7107 .5370 .4817 .3375 .2905 .2122 .1670 .1179 .0545 .0010 Title 42-8 — a part of PROCEEDINGS, AMERICAN CONCRETE INSTITUTE Vol. 42 JOURNAL of the AMERICAN CONCRETE INSTITUTE (copyrighted) Vol. 17 No. 4 7400 SECOND BOULEVARD, DETROIT 2, MICHIGAN February 1946 Shrinkage Stresses in Concrete* By GERALD PICKETTf Member American Concrete Institute PART 2-APPLICATION OF THE THEORY PRESENTED IN PART 1 TO EXPERIMENTAL RESULTS Carlson's results on prisms drying from one end As mentioned in Part 1, Carlson 1 applied diffusion principles to the problem of computing both loss of moisture and distribution of shrinkage. The fundamental equations on which his computations were based are the equations to which Equations 5 and 22 of Part 1 reduce when the parameter B is set equal to infinity. In his experimental work the prisms were allowed to dry through one end only, the rest of the surface being sealed. Measurements were made over gage lines that were parallel to the direction of flow of moisture. These conditions appear to be most favorable for the direct measurement of the distribution of shrinkage tendency since in an unrestrained specimen shrinkage stresses should not have any appreciable effect on the unit shortening in the direction of moisture flow. In Fig. 3 of his paper Carlson showed two diagrams. One diagram gave the distribution of shrinkage as measured after a definite period of drying and the other gave the computed "distribution of drying" (loss of moisture) for different assumed coefficients of diffusion for the m period of drying. The observed distribution of shrinkage and the com- puted "distribution of drying" are in good agreement when the proper coefficient is selected. However, as shown by Fig. 1 of his paper, the measured loss in weight was not in very good agreement with the theory. Carlson could have obtained slightly better agreement between theory and measured shrinkage if he had taken surface conditions into account, ♦Part 1 of this paper was published in the ACI Journal, January. 1946, and includes (p. 194) the com- Pl VPr;5e^or e o f f Applied Mechanics. Kansas State College, Manhattan, Kan., formerly Portland Cement ition Research Laboratory, Chicago. (361) 362 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE February 1946 --rS - ^> ■ - -- - .-« - <y _\ - f5 _— r ---_--r-.-_- : m =zaoox/a1 f=a07/n./cs - oerd&y - -£ -*r - cemeni stfiaa - _■ - f Fig. 1 5— Comporiscn of observed and calculated course of shrinkage i.e., used a finite value for the parameter B. However, had he done so, the discrepancy between theory and measured loss in weight would have been greater than that shown. Carlson's work is important evidence in support of the hypothesis that shrinkage of concrete approximately follows the laws of diffusion. Shrinkage of prisms of various sizes drying from one or more sides In the work done in this laboratory measurements were made on gage lengths transverse to the direction of moisture flow. Since variations in shrinkage along the path of moisture flow result in stresses transverse to the direction of flow, the measurements include the strains produced h J These however, the specimens are long compared to their dimension m the direction of moisture flow and the stress-strain relation is linear, then as shown in Part I the shortening of the central axi< will be equal to the average shrinkage. The results to be discussed provide a test of the theory for conditions in which both size of specimen and number i j es are variable. the unit shortening versus davs of drving for three different sizes of prism* of the same mix and for the three different drying conditions discussed in Part 1. .Mix A and cement M, described in the Appendix, were used. The specimens were cured seven davs under water. Each point is the average of the results from two pn~ lne curves were constructed from computations based on the theoretical equations developed in Part 1. These equations, which give the theoreti- SHRINKAGE STRESSES IN CONCRETE 363 unit shortening = S a cal relationship between unit shortening, the constants of the material, and dimensions of the specimen have the form \ k ¥ bf where S a> is ultimate shrinkage for the assumed final drying, / is the surface factor, k is the diffusivity factor, and b and c are dimensions of the specimen. The exact form of the function, especially the way in which c/b enters into it, differs with the drying condition. The three constants £«,, /, and k were evaluated from average ex- perimental values for the pair of prisms of 2-in. square cross-section, drying from four exposed sides. From these same constants the curves were constructed, as shown in Fig. 15, not only for this pair but also for the theoretical unit shortening of the other eight pairs of prisms. The agreement between the experimental values and the calculated curves is fairly satisfactory except for two pairs of 3x3-in. specimens, which were observed to have cracked during drying and therefore could not be expected to shorten in accordance with the theory. Discussion of the validity of the theory on the basis of the foregoing data The data from those specimens that did not crack, together with the data given by Carlson, might seem to indicate rather conclusively that shrinkage does take place in accordance with the theory develops I in Part 1. However, such a conclusion would not be justified. A good fit between an equation and experimental data is necessary but it is not sufficient proof of a theory. Although constants in the equations of Part 1 may be chosen so that the theory given there will be in good agree- ment with experiment for certain measurements on spc.imens under a few different conditions, the theory should be expected to fail under some other conditions since it rests on some assumptions that are not wholly correct. Shrinkage is not linearly related to change in moisture content; the flow of moisture in concrete does not follow the law of diffusion; and the stress-strain relation is not linear. Since the assump- tions are not wholly correct, the factors S «> , /, k, and E that are supposed to characterize the material must be empirical, and experimentally determined numerical values of these factors will be different for different Tests on the same material. The good agreement between theory and the experimentally determined contraction of the specimens discussed above must be the result of the balancing of opposing effects. They u ill not necessarily balance the same way in another test. The foregoing criticism means that however promising the theory may appear from the results of a few experiments, the application of the theorv must be limited and extrapolation of the results to sizes of speci- mens or conditions of drying other than those for which the constants 364 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE February 1946 were determined cannot be made with confidence. The selection of •r given conditions constitutes the chief difficulty for the prad of the equations. This doee not mean that the theory is of little I considerable value. Tnn l equations are not rigorously correct and the *ants cai the meaning attached to them. But the linkage does follow the diffusion equation approxi- mately and that the defn a are approximately those gi the theoretical equations if the empirical constants selected to give fair agreement with experimental result-. Although the experimental results shown in Fig. 15 appear to be in good agreement with the theory, a close study shows the following in regard to those pri~ og from only one side: I 1 After pro -i of the prisms drying from one fflde I - • ^ indicated by the theoretical curv< - --:u. prisms deviate more from •' Lxl-in The lxl-in. the same lvalues as the 2x2-in. -till more I - -- - tion. I- Lid result in still greater di>- omputed values unless allow] th change in lvalue. Comparison of dota on warping with data on shortening ded by the results shown in •.re root eriod ! the prism.* As mentioned in is The abscises give* a nearly straight i entering and warping. ■ the thickness of the specimen puts able basis. Multiplying the -- span by the thickness put* the on the same basis in regard to unit defoi J j l - 15 that were drying from only - trying fn - OS. iiammc* in the dmiio: kn-iu SHRINKAGE STRESSES IN CONCRETE 365 4000 V) JZ -t— c o V)00 b c en c 2000 c 0) I— o x: 1000 -»- c 3 <^ in .40 <U c r & S.30 r\, v> *) <L> E t) t— £.20 c h o <ij ^ < u tj dj .C: Q) *.10 o Ci t— a OJ 0- h f QJ * CJ Points are Experimental I Curves are Theoretical, based on /* - o.o7-in. per day — k -a 0J5 -/a ? p £T flto^" 12 10 8 • Experimental for 2 -in Prism o « " /-/A I [^ - JoAC curves \f i u o. 07 -in. per day ■ 0.035 -in. 2 • " f^oo -2ff00*/0~ tf Dashed curves if *o. 032 -in per day \k * o.o zo -in. 2 u Z 4 6 8 10 12 Square Root of Dans Drying _ Vt~ Thickness fn inches b Fig. 16 Comparison of observed and calculated course in shortening and warping of prisms of mix A and cement M The curves in the upper diagram and the solid d the i diagram were constructed from the theoretical equations using the same values of the constants S », /, and k as were used in construe the curves of Fig. 15. Therefore, the three curves in the upper diau of Kg. 16 represent the same equations as three of the curves in Fig. 15; they differ only in the abscissas. The solid curves in the lo jram deviate considerably from the plotted point-. This indicate greater disagreement with theory than is shown in Kg. 15. II the dashed curves obtained from the theoretical equation by using the Bame valtie of Sea but with / reduced by 54 per cent and k reduced by 43 per cent are in very good agreement with the experimental values. 366 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE February 1946 The computed shrinkage stresses will be about the same whether the first or the reduced values of/ and k are used. The values of the constants £», /, and fc, used in constructing the dashed curves, were obtained from three measurements as follows: (1) maximum warping of the 1-in. specimen, (2) time at which this maximum warping occurred, and (3) final shortening of a companion specimen. The agreement throughout the course of drying between the experi- mental values for warping and those given by the theoretical equations when these constants are used is excellent for the 1-in. specimen and very good for the 2-in. specimen. The above shows that if data on warping and data on shortening are analyzed separately, either group of data will appeal- to be in accord with the theory if the thicknesses of the specimens do not differ too much, but the values of/ and k obtained from the two groups of data will be different. The fact that the factor k is an empirical rather than a fun- damental property of the material is believed to be the chief reason why both groups of di * nted satisfactorily by on< nits. The empirical nature off is considered to be of only e in this study because it has much less effect than ■time arid warping-vs.-time relations. Effect of differences in k on warping and shortening diffusivity on the theoretical value- of ping (Equ 2 bortening (Equation 2:>> : are shown in Fig. 17 v ' plotted against the parameter V// jTfor ■ ■■■ fc. As shown, differences in k have ply warping; each curve follows the same maximum point. The lower k t the greater m maximum value of warping, of ticipated because of the effect of k through tb As shown by the curves for I upon shortening is entirely different - The rate of shortening is materially reduced n the maximum shortening is unaffecfc i rent effects of changes in k on irping has been useful in explaining differences in made with cements of different composition. I results from concretes made with two diff- - - Mix C was used. A comparison of 7 e conclusion that the coefficient of di- nt No. 5-1500-1.9 IS loner than for con- SHRINKAGE STRESSES IN CONCRETE 367 cretes made with cement No. 1-1500-1.9*. Concretes made with cement from clinker No. 5 shortened at a lower rate but according to data not plotted eventually shortened more than concretes from clinker No. 1. Before this explanation was found, it seemed surprising that of two groups of specimens drying from one side only, subjected to the same exposure, one group would warp more and shorten less than the other group. In order that one specimen warp less than a second when the two specimens have the same average shrinkage, the distribution of shrinkage in the first specimen would have to be more nearly uniform. For the same surface conditions, a large value of k through the para- meter/6 k, tends to make shrinkage more nearly uniform and therefore is accompanied by less warping. An increase in uniformity of shrink- age also reduces the shrinkage stresses in an unrestrained specimen and therefore reduces the tendency for spontaneous cracking. (Fig. 14 Part 1 — shows how the theoretical maximum stresses depend on the parameter fb/k (= B) .) Effect of alkali content on k and its possible effects on cracking It had been observed from various laboratory tests designed to measure cracking tendencies that concretes made with cements from clinker No. 5 tended to crack more than those made with cement from clinker No. 1, even though measurements often showed less volume change at the end of a given period of drying for the concretes of clinker No. 5. This greater cracking tendency of cement from clinker No. 5 was at- tributed to its higher alkali content, since this appeared to be the only important difference in their chemical compositions. Attempts to evaluate k for concretes made with cements from these two clinkers showed that for the same mix proportions the value of k for concrete made with the cement of higher-alkali content was only one-half that made with the cement of lower-alkali content. These observations sug- gested the possibility that: alkali reduced k, a reduced k resulted in higher shrinkage stresses, and higher stresses resulted in more cracking. To investigate this effect of differences in alkali content more fully, several tests were made using cement No. 1-1665-2.48. The procedure was to add 0.91 per cent Na 2 by weight of cement in the form of NaOH to the mixing water of one of two companion mixes. The results of one using mix B are shown in Fig. 19 where shortening and weight losses of prisms are plotted against period of drying. The dimensions of the prisms were 2^x23^x1134 in. They dried from all surfaces except the ends. By using for the specimens containing added alkali a time-scale equal to one-third the scale used for the regular specimens the corresponding curves for both sets of specimens approxi- *As explained in the Appendix, the first number is the clinker number, the second is the specific surface | Wagnea mel hod), and the third is the percentage of S0 3 . 368 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE February 1946 Fig. 18— Warp- ing and shorten- ing of prisms that differ primarily in the alkali con- tent of th« ment used Prisms 3-in. thick in the direction of mois- ture travel. Deflec- tion measured over a 32-in span. Mix C. Cured 7 days. Fig. 17 — Theoreti- cal effect of k on the course of con- traction and warp- ing of prisms dry- ing from one side only 4 6 8 10 12 I4 le 2 4 6 8 10 Square Root of Number of Days Drying 20 30 40 50 60 Period of Drying -days (without added alkali) 6C 90 120 ISO I6C ?io Penod of Drymg-days (alkali added) Fig. 19— Shrink- age and weight- loss for speci- mens with and without added alkali SHRINKAGE STRESSES IN CONCRETE 369 mately coincided, indicating that the main effect of the added alkali was to reduce the diffusivities for both shrinkage and moisture flow to one-third the value without added alkali. The effect of added alkali in reducing the diffusivity of shrinkage for cement of clinker No. 1 is in accord with data reported by Haeger- mann. 21 Haegermann was primarily interested in the effects on shrink- age of additions of various sulfates to cements of different C 3 A contents. The sulfates tried were ferrous, calcium, magnesium, sodium, and po- tassium. The amounts added were such as to increase the S0 3 , content 1 per cent, based on the cement. Five cements ranging from 15 per cent computed C Z A content to zero per cent C 3 A were investigated. The data were presented by Haegermann in the form of curves. For each cement, the curves representing the sodium and potassium sulfate additions are of noticeably different shape from the other curves for the same cement, the difference in shape being such as would result from a lower diffusivity. Since Haegermann did not give data on loss in weight during drying, it can only be inferred from the data on shrinkage that the sodium and potassium sulfates also reduced the diffusivity of moisture flow. From theoretical consideration, it appears that any highly soluble material should reduce the relative rate of drying; i.e., should increase the time required to lose a given percentage of the total amount of moisture to be lost.* However, since many other factors affect the rate of shrinkage, and alkalies have many other effects which may in- directly affect shrinkage, one should expect many real and apparent contradictions to the above indication that an increase in alkali content will retard shrinkage. The effect of the added alkali on cracking was investigated by means of the "wedge test"t and by the " restrained-shrinkage test" (subse- quently described). The result was that specimens of higher alkali content showed a much greater tendency to crack, as measured by these tests. Other tests made in this laboratory show that for cements containing an appreciable percentage of tricalcium aluminate, an increase in alkali content will increase final shrinkage of laboratory specimens unless the increase in alkali is accompanied by an increase in gypsum. The greater ♦This reasoning is based on the supposition that at least part of the flow of water in concrete is by means of the following cycle: evaporation at an air-water interface, vapor diffusion across air space, capillary flow ?n feJdfSEd Space and again evaporation at air-water interface. .Since the diffusivity of the soluble ma- terial within the liquid is finite rather than infinite, at any air-water interface at which water is evaporating ISL of soluble material will be higher than that for equilibrium with the adjacent liquid and therebv tend to restrict evaporation at this interface, and at any air-water interface at which water is con- densing the conrei.tration of soluble material will be lower than that for equilibrium with the adjacent Houid and therebv ten<l to restrict condensation at this interface. Therefore any highly soluble material should retard the'drving bv reducing the diffu-ivity of moisture flow. .„.,-.,. „ , t A speninei! L cast 5f1 he form of a wedge and. after curing, is permitted to dry from the two non-parallel surfaces. 370 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE February 1946 tendency to crack of the specimens with higher alkali content might have been due, at least in part, to a decrease in diffusivity and an in- crease in final shrinkage. On the other hand, the possible benefits from alkali should not be overlooked. The lowered rate of moisture loss will permit the interior of concrete to retain sufficient moisture for additional hydration for a longer time after drying of the surface begins. Prevention of complete drying of the interior during the usual drying season should be especially ad- vantageous in preventing cracking when restraints against shortening are present. Tests in this laboratory have also shown that concretes of higher alkali content have greater capacity for plastic flow, which is a favorable property. Decrease in k as drying proceeds As drying proceeds, the value of the coefficient of shrinkage diffusivity t apparently decreases. This decrease no doubt results from a pro- gressive decrease in the apparent diffusivity of moisture, diffusivity of moisture probably being a function of the moisture content. If diffusi- vity of moisture is a function of moisture content, then the shrinkage diffusivity can be considered to be a function of the shrinkage S and the differential equation becomes non-linear. Adding particular solutions, as was done in Part 1, is then not permissible. However, if in place of considering k to be a function of the dependent variable S it is considered to be a function of the independent variable t and of the dimensions of the body, then the differential equation re- mains linear. Furthermore, if the factor/is considered to varv with time in a hke manner so that the ratio f/k remains constant (see, for example lation 2a), then all of the equations for displacements, stresses and ' - developed in Part 1 still apply if the symbol t appearing in then, d by a function of t and the dimensions. The changes suggested •■'iH.ve amount to a , ontinual change in the time-scale so that the time squired for given conditions to develop becomes progressively longer - ,n,,dl,vl "« the theory in this way better agreement with experimental results can he obtained. Fig. 20 is an example of applying the foregoing analysis. The plotted points are from experimental data on the average warping of four 3-in specimens ol concrete of mix B with cement 1-2280-]. 94. When an attempt „ras made to select constant values of/, Ife, and .<J. to be used "' "■ theoretical equation that would give curves in agreement w,.h all -•> the experimental values, no( all 1he (1:((a ( , )ljl(i 1)( , bK)ughl ^ men wnh the theoretical equation. But by taking the following lor the factors, a Letter fit was obtained. SHRINKAGE STRESSES IN CONCRETE 371 0.06 0-02 n nr\ a 2 3 4 S 6 Square Root of Dags Drying VT Thickness in inches b Fig. 20 Com- parison of theo- retical and ex- perimental warp- ing Points are from the average warping of four 3 - i n beams. Span 32-inches. Mix B. Cement 1-2280 1.94. Cured 7 days. ft = 0.10-J^j^inVday ■fh f = 1.67 ft in dav, i.e., - = 5 k Soo = 765 x 10' 6 When these values are introduced into the differential equations and a solution made, the symbol T in the final equations for warping, etc., is replaced by ±k k [ 2 + t 1 b 2 L^ 2 where k is the initial value of k or 0.10 sq. in. per day. For convenience in making computations preliminary to plotting of the theoretical curve, b 2 T b A T 2 t was expressed in terms of T, or t = 1 . The tabular values used for constructing the curve are given below : From Table 4 Computed Values 2bv max <t T 3M» i b V max b 0.01 0.0152 1.00 0.333 0.0178 0.03 0.0326 3.61 0.635 0.0384 0.10 0.0538 li). 1 1.46 0.0634 0.15 0.0557 36.4 2.01 0.0655 0.20 0.0539 58.5 2.55 0.0635 0.30 0.0470 118 3.62 0.0553 0.50 0.0336 298 5.75 0.03 0.75 0.021U 639 8.41 0.0258 The better agreement that can be obtained by the modified theory probably would not compensate for the extra work in all cases. Since the 372 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE February 1946 Fig 22 Shrink- oge of slabs of n e s s • s drying from two oppo- des Fig. 21 Com- parison of theo- retical and ex- perimental warp- ing of beam ab- sorbing moisture from one side Poinli MPMHl th« warping ol a 1 -in b»om, having all but on* of i»t *urfac*s »»ol»d, during »ub- mtnion in water Cured 7 da yi and rh»n dn#d at bO R H for 10 month • b*for« Ivilmg Mi* 6 C»m»nl 1-9980 1 94 Span3Jinchei ' .l.|.- the valm Worp,ng ol prisms during absorption of moisture from one s.de SHRINKAGE STRESSES IN CONCRETE 373 zero warp where it remained for the rest of the test, a period of one month. The experimental value for the warp after 6 hours' exposure is considered to be in error. Other tests on 1-in. prisms did not show the indicated large decrease in warp between the fourth and sixth hours of exposure. Although the amount of experimental data on swelling is yet small, the indications are that the application of the theory as developed in Part 1 is limited, first, because at the beginning of wetting the moisture already present will ordinarily not be uniformly distributed; second, because of having remained wet longer, the cement in the interior re- gions will have hydrated more than that closer to the drying surface; and third, as the concrete becomes wet again, hydration again starts. Agreement with diffusion theory is not expected while hydration is occurring at an appreciable rate, especially if the formation of hydration products causes expansion. Effect of thickness on rate and amount of shrinkage of walls or slabs In an investigation of the effect of wall or slab thickness on the rate and amount of shrinkage the results shown in Fig. 22 were obtained. The specimens from which the data were taken were made of mix B. Cements of two different compositions and a fine and a coarse grind of each are represented. The specimens were cured seven days under water. The specimens were 34 inches long and of either 2x6- or 2xl2-in. cross section. By sealing all but two surfaces the prisms were made to represent slabs or walls of 2-, 6-, and 12-in. thicknesses drying from two opposite sides. For example, the specimens that represented a wall 12 in. thick were 2xl2x34-in. and dried from only the 2x34-in. surfaces, there being 12 inches between these surfaces. Sets of gage-points were cast in these pseudo slabs so that the short- ening over three or four 30-in, parallel gage lines could be measured on each specimen. Details are shown in the Appendix. Each curve was obtained by averaging the results from four specimens of a kind. As shown in Fig. 22, the results from these concrete specimens are in general similar to those obtained on the cement-silica mix discussed previously (see upper diagram, Fig. 16.). The curves have the charac- teristic S-shape found for similar plotting of data from smaller specimens. The thicker the slab the greater its fb/k ( = B) and, according to theory as shown by Fig. 8 (of Part I), jthe greater the shortening should be for a given value of the abscissa, ^t/b. The experimental data are partly in agreement and partly in disagreement with the theory in this regard. In the middle, straight-line portions of the curves, the curves are in the correct positions relative to each other, but in ever y case the relative positions become reversed at larger values of Vf/6. Also, the relative 374 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE February 1946 presenting the two < rinds are rev< - portion of the curves. This latter deviation on the basis of a non-linear stress-flow relation- - the negative, the specimen will >uld if plastic Bon did not take place. Of and negative Bows are equal (algebraic the specimen is not changed by plastic Bon in an unrestrained Bpecimen if the However, if the stress-flo* relation- bhe range of stresses developed, the than the first power of th< o( tin specimen is reduced by pi pronounced foi the thick< I Jurve H of Fig i i I the '!..• basu of a aon-lin< ol the Brat p rind* wei -„,i enti rig the I othei teste with Plorf.c How • SHRINKAGE STRESSES IN CONCRETE 375 In the usual measurements of plastic flow the quantity measured is the inelastic deformation of a body that results from applied loads. From these measurements computations are made of the inelastic defor- mations of the individual elements, i.e., average unit deformation if the load is axial or unit deformation of the outer fiber if the load pro- duces flexure. If stresses from other sources are not present, the com- puted values may be representative of the actual plastic flow. But if stresses from other sources are present, the computed and actual values may differ appreciably. Therefore, if in addition to load stresses a specimen is under stress as a result of non-uniform temperature or non- uniform shrinkage, it should be made clear whether the term plastic flow refers to the resultant plastic flows of elements or to only computed plastic Bows produced by loads. Since the effects of load and the effects of < Irving are not simply additive, there is no clear basis for deciding how much of the total deformation is due to the stresses arising direct!} from the load. In agreement with previous writers, the deformations produced by loads will be taken as the difference between the deforma- tions of loaded specimens and the deformations of identical specimens under the same drying conditions but not under load. Only the defor- mations produced by loads will be computed and n I ><\ i but in the interpretation of results consideration will be given to what the actual inelastic deformations are believed to As shown by the formulas for plastic flow used in this paper, the t deformation produced by load is divided into tv ~, elastic and inelastic. The elastic part is considered to be that which would be recovered immediately if the load were rein- ed it is determined the computed load-stresses and the "dynamic" modulus of elastii The remaining part is considered to be the plastic How produced by the load. Some investigators make a slightly different division in that the elastic deformation is considered to be that which was product diately upon application of the load rather than that which would recovered immediately upon removal of the load. The two \ equal if the modulus of elasticity does not change during the test. Some writers prefer to divide the total deformation produced by load into I parts: ( 1) that recovered immediately upon removal of load, (2) that not immediately but eventually recovered, (3) the permanent deformation. McIIcnrv -' restricts the use of the term plastic flow to the third part. This division into three parts has merit, especially for those which the second part is an appreciable percentage of the total. For the data given in this paper no separation of the second and third parts could be made, but the permanent deformation (3) is believed to be much greater than the temporary (2). 376 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE February 1946 If the specimen is not shrinking while it is under load, then a con- siderable part of the inelastic deformation is probably only temporary and apparently the result of viscous flow in the adsorbed water films. After removal of the load, the elastic constituents of the gel-structure tend to restore the original shape but are retarded by the viscosity of the adsorbed water films. However, if an elemental volume* of cement paste is shrinking while under stress, the conditions are different. The loss of moisture introduces relatively large interparticle forces which tend to change the relative positions of the colloidal particles within an element. Some adjacent particles are pulled closer together but others are moved further apart. During this time of movement the directions of relative motion of the particles may be appreciably affected by stn on the element. In this v ses on an element during the time it is shrinking may produce comparatively large permanent deformations. The foregoing is one explanation for the much larger amount of plastic flow that a load will produce on a drying specimen compared to what it would produce if either tin- specimen were prevented from drying or had pr< been dried. It is also an explanation of the relatively • near the drying surface to deform plasti- cally without cracking. I) the analysis is correct, then a definite stn — flow relationship cannot l»< given element of concrete since vould depend not only upon the magnitude duration i on an element but also upon the changes in mois- d while the element was under stress. formation was considered to be per- i in deformation with time was con- hiefiy by changes in distribution of shrinl. i ly small lag in time after the develop- !n "pla8tic Mow" rather than "creep" ;>.i! t of t he deformation. hat way the pa-tic flows of the individual deformations ol the body as a whole. dal load the plastic flow in ten-ion or . the load is the difference in the a um [nation ol each dement and what the algebraic body bad not been under load. But the pis ral load depends on the moment of the inela ■ lement with neutral axu " Both aon side and plastic compression on the .1 load contribute to the measured ' SHRINKAGE STRESSES IN CONCRETE 377 Summary of remarks on plastic flow. The actual plastic deformation of elemental volumes of a specimen may be much different from that computed on the basis of laboratory experiments if shrinkage-stresses are present, but in this paper the plotted curves represent such computed values. Computed values are based upon the difference in the deforma- tions of loaded and not loaded specimens. The term "plastic flow" is used in this paper to refer to either actual or computed plastic deforma- tion. Plastic deformation is arbitrarily defined as that part of the total deformation produced by stress (either by actual stress or by load-stress as indicated by the text) that would not be immediately recovered upon removal of the stress. Effect of thickness on stresses and plastic Flow when the slab is partially restrained against shortening Companion specimens of the same size and sealed in the same manner as those represented in Fig. 22 were partially restrained against shrink- age by specially designed steel bars, somewhat as were those described by Carlson. 23 The main features of the steel bars are shown in Fig. 23. (The concrete specimen illustrated in Fig. 23, however, is from another test in which the concrete was allowed to dry from all sides and only one bar was used per specimen). Each specimen of 2x6-in. cross section con- tained two %-m. diameter bars, and each specimen of 2xl2-in. cross section contained four 5^-in. diameter bars. The arrangement of bars is shown in the Appendix, and in Fig. 24. A rubber tube covered the central 20 inches of each bar so as to prevent bond over a 20-in. gage length, thereby insuring the same axial force in the bar over all sections of the gage length. That part of each steel bar not covered with rubber was threaded and thus the bars were anchored to the concrete for a distance of 7 in. on each side of the gage length. Be- cause of this anchorage the shortening of the steel bar over the gage length is equal to the shortening of the concrete over the same gage length. Moreover, as is obvious from considerations of equilibrium, the force in the concrete in this gage length is equal and opposite to the force in the steel in the same gage length. Therefore, the average unit stress in the concrete can be computed from the change in length, modu- lus of elasticity, and percentage of steel. The formula is A S E S M (Jr = A c I where <x c is average stress in the concrete A 9 is cross-sectional area of the steel A c is cross-sectional area of the concrete E 8 is Young's modulus for the steel 378 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE February 1946 n ( dUb -vHHIHHK • [ fS c 01 E u V a (/) r c N CO 5 1 i W \ \ v 1 ^r -0 9 — 9-*T _ * 8 ° S — Mr i SP«Z P/,;# ■ V/ £ 2Qn± J*qqn# yjoouj? i = i\ o *■_ «t > I <3 ■ it c . .'C ■z iz 9- SHRINKAGE STRESSES IN CONCRETE 379 A/ is net change in length over gage length after corrections have been made for any change in temperature.* id is nega- tive if the specimen has shortened. / is gage length. Not only the average stress caused by the restraint but also plastic flow caused by this stress can be computed if the modulus of elasticity of the concrete is known and the assumption is made that the shrinkage tendencies of the restrained specimens are the same as those for com- panion unrestrained specimens of the same size. The formula is ° = S " -B. + 1 where c is unit plastic flow caused by restraint, S av is the unit shortening of the free-shrinkage specimens, and E c is Young's modulus for the concrete. Performance of partially restrained specimens. As explained in the Appendix all specimens were cured under water. The specimens tended to expand during this storage and consequently the concrete in those partially restrained with restraining bars was compressed. Therefore, for a short time after drying began, the direction of the plastic flow produced by the restraining bars was in a negative direction. Shortly after drying began, the stresses in the restraining bars changed from tensile to compressive, and the average stress in the concrete changed from compressive to tensile. Under the conditions of this test the average stress reaches a maximum and then slowly decreases if failure by spontaneous cracking does not occur. A specimen's average stress and its shortening necessarily reach their maximums simultaneously if the temperature remains constant. Therefore, the time of maximum average stress is the time when the rate of average shrinkage equals the rate of plastic deformation. During the decrease of average stress, the rate of plastic deformation exceeds the rate of shrinking. Ordinarily in this test the specimens are not permitted to reach a final equilibrium state in regard to shrinkage, shrinkage-stress, and plastic flow. But just after the maximum restraining force has been developed additional tensile load sufficient to cause failure of the specimen is applied. This load is applied to the protruding threaded ends of the restraining bars by a machine designed for the purpose. While the load is being applied, measurements are taken so that the added stress in the concrete can be determined. Further details are given in Fig. 33 of the Appendix. ♦All tests were conducted in a room maintained at 76 =±= 1°F and a relative humidity of 50 =•= 2%, except for occasional deviations from these limits. # 380 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE February 1946 500, , m Fig. 25— Plastic flow in the par- Hall/ restrained specimens of Fig. 24 Cement 1-2260 -/. 94 Factor of safety. The purpose of the testing just described is to learn how close the specimen comes to cracking spontaneously The ratio of the computed stress at failure to the maximum average shrinkage- stress is called a factor of safety. Specimens that crack spontaneously are reported as having a factor of safety Less than unitv. Results showing computed average shrinkage-stresses and plastic deformation are shown in Fig 24 and 25, respectively. The record of the number of specimens of each cement that cracked spontaneously and of the average factors of safety (f.s.) of those that did not crack is also shown in Fig. 24. Effect of thickness on plastic flow. Attention is called to the similarity ol the three sets of curves in Fig. 22, 24, and 25. The similarity is not to be interpreted warily indicating that the plastic deformation is proportional to its stress. One is tempted to make this 1 " t " , , , 1 "V e » « it were true, then the plastic flow of a specimen would depend only on average si ress and not on the distribution of stress In general, the diagrams show that the 12-in. specimens developed con- siderably more plastic flow for the same amount of average stress than «<* -- or the 6-in. specimens. T1 >i possible reasons fortius: first, because of the lower temng, the larger specimens will have been under a given - longer than the smaller specimens and therefore would be expected to have more plastic flow for the same stress. Since the time ^quired for the same amount of shortening is approximately pro, " i:i <" "'<• 7— - the thickness, the 12-in. specimens will in genial have been under a given range of stress about four times as long as the lH ": ~»™- Second, sin,,, the thicker specimens will have higher maxnnum etresses, the additional plastic flow could be accounted for by ■ non-hnear stress-flow relationship whether or not this reTtionsh* SHRINKAGE STRESSES IN CONCRETE 381 for each element was modified while the element was losing moisture rapidly. Third, the assumption that shrinkage tendencies of the re- strained-shrinkage and free-shrinkage specimens are equal is not entirely correct and consequently their computed plastic flow are in error. A difference in the shrinkage tendencies of the 12-in. free and restrained specimens might result since the arrangement of the four bars was such as partially to obstruct the flow of moisture. Probably all factors listed above contributed to the results. Of the factors causing the computed plastic flow to be greater in the 12-in. specimens, the author is of the opinion that the non-linear stress-flow relation contributed much more than the difference in duration of given stresses.* That the maximum stresses in the larger specimens are higher is shown by the fact that all the 12-in. restrained specimens of three of the four cements cracked (see Fig. 24) whereas only a few of the 6-in. specimens and none of the 2-in. specimens cracked. According to most of these arguments the 2-in. specimens should have less plastic flow than the 6-in. specimens, whereas in general they have slightly more for the same shortening and for the same average stress. A complete explanation for this is not at hand, but the lesser extent of hydration of the cement in the 2-in. specimens because of their more rapid drying may be a factor. Also, the exposed surfaces of the 2-in. specimens were the top and bottom surfaces as cast, whereas the drying surfaces for all the other specimens were the sides as cast. Bleed- ing and settlement of the plastic mix before initial hardening plete always makes the concrete near the top and that near the bottom as cast different from that at the sides. It must also be remembered that the computed plastic deformation may be more or less than the real plastic deformation of the material. Use of beams drying from only one side for determining probable stresses in slabs or walls drying from two opposite sides To obtain information on plastic flow and on the magnitude and distribution of stresses in unrestrained walls or slabs drying from two opposite sides another set of specimens, also companion to those repre- sented in Fig. 22, were made. These specimens differed from those of Fig. 22 in that they were permitted to dry from only one side in- of two opposite sides and in that the thicknesses of corresponding speci- mens were just half those of Fig. 22. Since they were half as thick and dried from only one side instead of two sides (see Appendix), any one of se specimens was considered to have the same conditions of di and consequently the same distribution of shrinkage tendency as either half of a corresponding specimen represented in Fig. 22. ♦Most contemporary writers on the inelastic properties of concrete apparently would take the opposite view. ThS difference "in viewpoint » explained and an argument for the author's view is given in Ref. 20. 382 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE February 1946 If the distribution of shrinkage tendency is the same and if the correct external forces are applied so as to make all the deformations the same as those of either half of the corresponding specimen, then the distribu- tion of stresses will also be the same. Eight specimens of a kind were made, four of which were allowed to warp freely and four were restrained against warping. It was not feasible to distribute the external restraining forces on the specimens to be restrained against warping in exactly the manner that the mutual forces between the two halves of the corresponding specimens were ibuted. As shown in the Appendix, the method adopted ws support the specimen as a simple beam and to apply enough force at the quarter-points to prevent warping of the central half. Ifi discussed previously and as shown by Equation 19*, the amount of warping of a specimen free to warp is indicative of the non-uniformity linkage tendency. From similar considerations it follows that the sary to prevent warping is indicative of the non- uniformil ee resulting from the non-uniformity of shrinkage Furtli. the difference between the actual moment required ing and that computed from the amount of flow, gives an indication of the dis- tribution f ] one of the four cements are give the actual moment developed, livided I ection modulus, / c, for beams of three different what the Mc I would have been ment had not produced plastic flow. The upper measured values of the warping tendency and x ; panioo srx cimens. Foung's modulus was 1 quency of vibration. ^ ace the ordinatee in Pig. 26 are i at the stress in the outer fiber according Formula. According to the lower curves of or the 1 -in. specimens reached B maximum kefirs* day ol drying. Thecomputed 3-in specimen was 190 lb. per sq. in. and mputed Btreas in the 6-in. specimens had and the indications are that, had the uued, the computed beds maximum • r about LOO days ol drying. om- reon the! The acta ■ the drj rapidly SHRINKAGE STRESSES IN CONCRETE 383 Fig. 26 — Com- parison of actual moment neces- sary to keep beams from warping with the conputed mo- ment necessary to strai g hten companion beams that are free to warp. (Plotted in terms will build up as rapidly in the thicker slabs as in the thinner. The actual stresses are probably better represented by the solid curves of Kg. 27. Stresses based on modified theory. The solid curves of Fig. 27 show- stresses based upon a modified theory. These curves were obtained by substituting appropriate values of the parameters y/b, kt/b 2 ,fb/k, E, and So into Equation 20 of Part 1. For the construction of these curves the theory as presented in Part 1 was modified in that, instead of using constant values for the factors k, f, Soo } and E, the following procedure was pursued: (1) The ultimate shrinkage S*> was set equal to 750, 700, and 600 millionths, respectively, for the 1-, 3-, and 6-in. thick specimens. The selection of these separate values rather than one value for all specimens was governed by the apparent ultimate unit shortenings of the cor- responding free-shrinkage specimens (Fig. 22). (2) The ratio f/k was set equal to 2.5 in." 1 , i.e., fb/k was 2.5 for 1-in., 7.5 for 3-in. and 15.0 for 6-in. specimens. When this value of f/k and the above values of &«, were used, the theoretical maximum values of warping as given by curve A of Fig. 14* were found to be in agreement with the experimental values of maximum warping for each of the three thicknesses of specimens. (3) A value of kt/b 2 was selected for each period of drying (1, 7, and 28 days) and for each thickness of specimen, such that when substituted along with the above values of Sa> and f/k in Equation 23* for warping the result would be in agreement with the experimentally determined values for these periods and these thicknesses. (4) A value of E was selected for each period of drying and each thick- ness of specimen such that the theoretical moment given by Equation 27 would be in agreement with the experimentally determined values. ♦See Part 1. 384 Fig. 27 Theo- retical distribu- tion of shrinkage stresses (modified theory) in slabs drying from two opposite surfaces for various dry- ing periods and slab thicknesses JOURNAL OF THE AMERICAN CONCRETE INSTITUTE 12 -in. Slab 6-in.5!ab W 2000 1500 1000k 500 -500 2000 1500 1000 500 -500 2000 i 1500 1000 500 After Drying 1 Day •Theoretical stress based on dynamic £ ^/Theoretical stress basea on reduced E > < [ ; I '_ \~ dynamic reduced J i \- dynamic After Drying 7 Days L— dynamic \ \ y* reduced A -500 L After Drying 28 Days dynamic reduced \T- ">ahnic reduced 2 3 4 S 6 12 3 inches from Exposed Surface February 1946 2-in.Slab — dynamic r r dynamic v ^-dynamic \ (Educed \ I 5 ' lues "• S , kt b\fb fc, and E together with appro- bstituted into Equation 20* and comp olid curves were plotted from bl sho* the stresses that arc obtained when the f< imputing is used except thai the dynamic E. The difference in th ua a by plastic How and thus is i .., !asti< deformation that baa occur* I thai for the most part the magnitude and es after the various | I drying and for the Besses ol specimens are about as given bj i ire probably in error in a Fheprin- "P* d error lies in the assumption thai the i aken Into account by using i reduced modulus of elasti - done when Equation 20 ends oo the- past stress-hisi mJv M •See J SHRINKAGE STRESSES IN CONCRETE 385 stress at the moment. Furthermore, as explained in the section on plastic flow, the rate of flow for an element will depend on the rate at which the element is tending to shrink and may not be proportional to the stress on the element. Therefore, near the drying surface, where the stress has been relatively high from the beginning of drying, the plastic flow will be greater and the stress will be less than that indicated. Slightly farther inward where the stress has only recently changed from compression to tension the resultant plastic deformation will be less and the stress more than that indicated. The dotted curve in the one diagram of Fig. 27 represents an attempt to show a better estimate of the actual stress. Reversal of stress by plastic flow. Fig. 26 indicates that eventually the stress in the outer fiber will become negative, i.e., compressive. In all restrained-warping tests that were continued until equilibrium of mois- ture content was nearly reached, the moment required to prevent warp- ing decreased to zero and would have then become negative if restraint against negative warping had been provided. This means that when a wall dries from two opposite sides or a prism dries from all four sides, eventually the outer shell will be in compression and the inner core will be in tension. Of interest in this connection is the fact that specimens of neat cement bars have been known to break spontaneously and audibly while resting in place in a storage rack. The explanation is that during the early part of drying large tensile stresses developed in the outer shell. As a result the outer shell was first permanently elongated and then caused to fail in tension, i.e., to crack. As drying proceeded inward, the inner core, which had not yet been stretched, tended to become shorter than the outer shell. The cracks closed, compression developed in the outer shell, and tension developed in the inner core. In some cases this tension was sufficient to cause failure of the core. A specimen would break spontaneously when failure of the inner core occurred at a section where the outer shell was already cracked. Investigation of properties of concrete by means of slabs or prisms drying from one side only As the foregoing has indicated, results from prisms drying from onl\ one side have been very valuable for ascertaining in what ways the theory of diffusion is applicable to shrinkage of concrete. They are also valuable for investigating certain properties of concrete, especially if used in connection with the diffusion theory. The chief advantage of drying a prism from only one side is that it tends to warp as well as shorten as it dries and thereby makes possible measurements not ob- tainable on prisms drying from all surfaces. 386 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE February 1946 As explained previously, the results from prisms drying from only one side indicated a cause for concretes of higher alkali content to show a greater tendency to crack under some conditions. Additional results in regard to fineness of grinding and percentage of gypsum will now be reported for the information they give to illustrate how such prisms may be used to investigate properties of concrete. Effect of finer grinding on plastic flow. As explained previously, in addition to the one cement represented in Fig. 2G and 27, three other cements were tested at the same time and in the same manner. The re- sults for the other cements were similar in most respects to those shown in Fig. 26 and 27 for the one cement. There were some differences, how- ever. More plastic flow occurred in the specimens made with the two finer-ground cements. The differences in plastic flow with fineness of grinding in tin- ' similar to and in agreement with the difl'er- bh( se cements in the restrained-shrinkage test (Fig. 25). The effect of finer grinding on strength is similar to that produced by r curing and therefore since longer curing decreases plastic flow we migh that fim ag would also decrease plastic flow. How- bed in this way all Bhowed that finer grinding stic flow for both restrained-shrinkage and restrained- m thifl laboratory indicate that this :mcr grinding on plastic fio\i is indirect. A given quantity of i i a1 the time ol grinding is less effective in retarding the earl} -l reactions the finer the cement. 11 Lack of proper re- tardation tctions because of insufficient gypsum results shrinkage tendency and greater capacity for ccording to the indications, finer grinding, if percentage of gypsum, indirectly produces tendency to deform inelastically. oswered the question introduced in the discussion by the i • plastic flow did not also reverse the the finer-ground cement I ' ibly at tl trying the coarser-ground cements, becaua gth, flow more readily bul become lees pis at the later ages when the strengths are • talized. I -nice only slight differences in the I viate from the theory could aceount fol d since i from the theory in many differ possible answers to the question. ited in Rg. 24, according to which grind Ited in u -e- at the end of the curing produced by the the SHRINKAGE STRESSES IN CONCRETE 387 concrete to expand during curing, especially the tendency to expand after some resistance to plastic flow had developed. Any tendency for the interior of the free-shrinkage specimens to continue expansion after the surface begins to dry would reduce the rate of shortening at the beginning of the drying period. If, as seems quite probable, this tendency is greatest for the thicker specimens with the coarser-ground cements, then these specimens would shorten relatively less at the beginning of drying than would be indicated by theory. Effect of added gypsum. The effect of the gypsum in the cement on the properties of the hardened concrete was observed in an investiga- tion in which 21 cements were made from the five clinker compositions listed in the Appendix. By blending various grinds of these clinkers, cements of different finenesses and different gypsum contents were obtained from each clinker. Concretes (Mix C) made from these 21 cements were tested in the manner indicated previously for prisms drying from only one side. However, in this investigation only the 3-in. size of specimen was used. Where the C^A content of the clinker was moderate or relatively high, an increase in SOz, content decreased shrinkage and warping and also decreased plastic flow. Where the C 3 A content was low, an increase in SOz had relatively little effect. According to other data obtained in this laboratory, a still further increase in SO z would have increased the shrinkage. 24 Representative results for a cement of high CzA content are shown in Fig. 28, 29, and 30. As shown by Fig. 28, the maximum warp of those specimens free to warp was reduced appreciably by in- crease in per cent of SOz- The reduction in shortening with increase in SOz agrees with that reported previously by other investigators 21 - 24 - 25 . Fig. 29 shows that the restraint developed by the specimens restrained against warping was in general less with the higher percentages of SOz- However, increasing the SOz from 1.5 to 2.4 per cent had only a very small effect on the amount of restraint developed in the restrained specimens compared to the effect on the warping of unrestrained speci- mens. The explanation is that although the increase in SOz reduced warping it also reduced the tendency to yield under stress. The net result is some reduction in stress but not as much as would be antici- pated from the results of the free-warping specimens. Fig. 30 shows the effect of SOz on the factor of safety against cracking as determined by this test of a cement of high CzA. SUMMARY AND CONCLUSIONS The theory that shrinkage of concrete follows the laws of diffusion similar to those followed by the flow of heat is tested by means of specially 388 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE February 1946 jba8| uo spunod - |uiej{S9^( siuuoiiiiai ui 6uiu8|j04? |iun 8 8 8 8 8 o 4° .1=' c — 5 * u * 0) .2 > I ^ Q. - >.£ 21- i ZE 2 5£ O ._ E > • o SHRINKAGE STRESSES IN CONCRETE 389 3.0 2.6 i ! r — i 1 — f r ■ t r r t 7 days) j ^ - Cured 7 days \ Jp. Surface 1500^^ ■ ~Zs^ X / 28 days - / s^ / ^^C/ 28 days ^^ \ 2IOO _ _.. II,,, i.i,]; ' Fig. 30 Effect of gypsum, peri- od of curing, and fineness of grind- ing on factor of safety against cracking Specimens same as those represented in Fig. 29. 10 1.5 2.0 2.5 3-0 3.5 5O3 Content of Cement, % by weight 4.0 designed experiments. According to the theory th developed in Part 1, the shrinking and development of stress in a given coni under given conditions of drying is consii be chai certain constants. These constants are diffusivity of shrinkage, suri factor, ultimate shrinkage, and Young's modulus of elasticity. Equa- tions were derived in Part 1 g I warping of prisma versus period of drying in terms of these constants and the dimensions of the prisms. In Part 2 it is shown that these con- m be seta shortening of a prism as computed by the theoretical equations 1- in good agreement with experimental values of shortening. Furthern it is >hou 11 that by using the same constants the shortening versus period of drying of other prisms differing in size and number of - to drying can be predicted with fair accuracy if the difference in si not too great. However, it is shown that the theory r modified to take into account inelastic deformation and to permit the supposed constants to vary with moisture id size of specimen if the theory is to be in agreement with all results on all t> mens of a given concrete. The theory is used to explain various things about concrete; in I paradoxically, it is used to explain some of th< which con< does not perform as predicted by tin- theory. The tendency of I specimens to crack more ami shrink Less than smaller specimens and the effect of alkali content of the cement in increasing the tendency to warp while reducing the rate of shrinkage are explained on th- the 390 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE February 1946 theory. It is shown that when a saturated specimen is dried at 50 per cent relative humidity the stress developed would be much greater than the strength of the concrete if it were not for the effects of plastic flow. It is further shown that when a specimen is restrained against deforming the restraining forces are much less than they would be if plastic flow did not occur. An example is given of the use of the theoretical equation in determin- ing the distribution of stresses at various times during the drying of a specimen. In this example, consideration is given to plastic flow and to the decrease in diffusivity of shrinkage as the specimen dries. The restrained-shrinkage test and the restrained-warping test are used to determine a factor of safety against cracking for concrete under conditions of drying and of restraint comparable to those under which the tests are made. These tests, together with tests on free-shrinkage and free-warping specimens, are used to measure plastic flow. The various tests described in Part 2 and in the Appendix to Part 2 when used in conjunction with the theory given in Part 1 provide a means for studying some of the more fundamental properties of concrete and for predicting the performance of concrete under some conditions in the field. ACKNOWLEDGMENTS This paper developed during a study of the causes and the control of cracking of concrete. The author is. indebted to many present and past workers in this field. E. A. Ripperger, now Lt. (j-g.) U.S.N, with the Pacific fleet, was responsible for a large part of the experimental work reported herein. He designed (or adapted from earlier designs) most of the special equipment used and was engaged in certain phases of the study previous to the author's participation. All of the work was done under the supervision of F. R. McMillan, Director of Research, and T. C. Powers, in charge of Basic Research. The author is particularly indebted to Mr. Powers for assistance in preparing the manuscripts. His suggestions in regard to presentation of material and the wording of various paragraphs have been invaluable. The author also wishes to thank Miss Adele Scott for preparing the diagrams and Miss Virginia Atherton for proofreading the manuscript. SHRINKAGE STRESSES IN CONCRETE 391 APPENDIX TO PART 2 Mix proportions Parts by Weight Mix A Mix B Mix C 0.5 0.5 1.0 0.6 0.355 to 0.388 0.048 0.403 to 0.436 1.0 1.28 1.82 487 Water added for absorption 083 570 1 Elgin sand 2 43 2 97 Consistency Consistency of Mix B with different cements was maintained fairly constant at from 5 to 6 in. of slump with a 12-in. cone by varying the amount of mixing water. The consistencies of Mixes A and C were allowed to vary with the different cements. Mix C usually gave a slump of from 2 to 4 inches, but with some cements the slump was as little as 1.5 inches and with others as much as 6 inches. Materials Cements: One cement designated M was a mixture of four brands of Type I cement, purchased in Chicago. Its specific surface by the Wagner method was 1665 sq. cm. per g. The other cements were prepared from five different commercial clinkers. From each of these clinkers cements of three different finenesses, coarse, medium, and fine, were prepared by grinding at the plant. In addition, two cements, one of low and one of high gypsum content, were prepared from each clinker by grinding in a small lab- oratory ball mill. The purpose in preparing these five different cements from each clinker was to make it possible to obtain any desired fineness and gypsum content by blending different grinds of the same clinker. In referring to these cements in the text the first number in the designation is the clinker number, the second is the Wagner specific surface, and the third is the per cent S0 3 content by weight. The chemical compositions of the five clinkers and of cement M are shown below. Oxides Cement Clinker No. Chemical Analysis, per cent by wt. (corr. for minor components) 20.67 23.05 27.82 5.48 4.14 1.93 2.50 4.35 1.87 65.00 64.28 65.38 1.31 1.36 1.75 0.19 0.03 0.17 0.85 1.05 0.26 2.71 0.73 0.23 0.30 0.05 0.05 0.40 0.17 0.22 Cement M SiOi AhOa... Fe20 2 Combined CaO MgO S0 3 Loss on Ign. . . . Free CaO Na 2 K 2 Compounds C3S C2* I \ < YU , 21.54 6.52 1.56 64 . 32 2,17 0.41 0.15 0.98 0.17 0.16 50.73 23.49 14.72 \ 75 22.56 5.00 2.48 64 06 3.35 0.20 0.37 nil 1.13 0.44 Computed Compound Composition per cent by wt. 66.57 52.37 38.58 51.61 9.05 26.58 50.66 25.75 10.29 3 61 1.95 9.06 7.61 13.24 5.69 7.55 21.25 5.98 2.69 62.56 3.04 1.75 1.13 0.79 0.28 0.63 44.15 27.62 11.30 ^ I'.i 392 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE February 1946 Aggregate: Sand and gravel were from Elgin, Illinois. The gravel was screened to pass a %-in, sieve and be retained on a Xo. 4 sieve. The sand was graded as folL Sieve Per Cent Xo. Bine 100 a 15 28 4:< 14 82 4 100 Pulverized Silica: The silica was from the same source as standard Ottawa sand, but ground to cement fineness. Its specific surface b)- the air-permeability method was 3200 sq. cm. per g. Procedure With some exceptions the procedure in preparing and testing the specimens was as follows: Preparing the specimens: The materials were mixed in a power-driven open-tub mixer. The fractions of the various-sized aggregates and grinds of cement were weighed and placed in the tub. The mixing schedule was: mix ] _> minute dry, 2 minutes wet, mix 2 minutes. (Special tests showed that the grinds of cement were adequately blended in }/> minute of dry mixing in the proence of sand.) The freshly mixed concrete was placed in steel molds and consolidated by light vibration by placing molds on a platform-type vibrator. Covers were fastened on the molds but not made water-tight. Each mold was equipped with restraining bars, gage inserts, etc., the details of assembly depending upon the tests to be made. The molds and contents were stored under water at 74 F for one day. The molds were then stripped and the specimens returned to water at 76 =«= 1 F in a covered tank where they were left for one hour. The specimens were then removed one at a time, dried with a cloth, and all initial measurements of length, deflection, and weight were made. All dimension measurements were made within 30 seconds after removal from water, and weighings were made as soon thereafter as practical. The specimens were then returned to the 76 F curing tank. One day before the end of the curing period the specimens were removed for sealing of certain surfaces against the loss of moisture. The surfaces to be sealed were wiped with a cloth and then allowed to air-dry until the surface just changed color. During .me. scheduled measuren. usually made. After the color change and before any appreciable loss of moisture by evaporation, one coat of black, quick-drying brushing lacquer was applied. After the lacquer had dried a few minutes, the - ■d with a cloth and one coat of hot paraffin was applied to the lacquered surface. While a second coat of paraffin was being applied, one thickness of t *- paper of appropriate shape wa- pressed into the still soft parafiin somewhat in the manner in which a paper hanger applies wallpaper. Next, a final heavy coat of paraffin ipplied. The layer of paper helped to eliminate pin holes. (When only the ends of prisms were Bealed, the paper and final coat were omitted.) r the specified surfa Jed. the unsealed surfaces and exposed steel parts were cleaned, vaseline was applied to the steel parts, and the specimen was returned to the curing tank for an additional day of curing. At the end of the curing period the specimens were transferred from the curing tank to a room maintained at 76 ± 1 F and SHRINKAGE STRESSES IN CONCRETE 393 50 ± 2 per cent relative humidity. At this time the vaseline was removed from the steel parts and measurements for the beginning of the drying period taken. Testing the specimens: The testing was considered to have begun in most of the tests with the beginning of the drying period. The specimens may be divided into classes, according to the tests made, as follows: Free shrinkage specimens were measured for length changes, weight losses and resonant frequency of vibration. Reference plugs were cast in the ends of the specimens for the length-change readings. They were hex-head cap screws arranged to give the desired gauge length. Ordinarily these were single plugs centrally located in the ends. But for those prisms that represented slabs drying from two opposite sides the arrange- ment was as shown in Fig. 31. o o o >i" \ii" \ti"\ f i" ] 6" -| o o o sea/ed^\ 6" h*H 1 Z" o 2i» o .__ J- o H' o 2* FH Fig. 31 — End views showing arrangement of gage plugs in specimens representing 2-in,, 6- in. and 12-in. slabs The gage plugs for these specimens were Y% in. bolts 4 in. long. Restrained shrinkage specimens were partially restrained against shrinkage by steel restraining bars. The arrangement of the bars in those specimens that represented slabs drying from two opposite sides was the same as that shown in Fig. 31 for gage plugs except that only two bars 3 inches apart were cast in the 2x6-in. specimens. Square specimens that are permitted to dry from all four sides and that are partially restrained by one centrally located bar are used in routine testing of resistance to cracking. Further details in regard to restraining bars and the measurements for change in length of the restraining bar are shown in Fig. 32 and 33 as well as in Fig. 23 body of the 1 1 If the restrained-shrinkage specimen did not crack spontaneously before the maximum restraining force had been developed, additional increments of load were applied as shown in Fig. 33 until failure was produced. Free warping specimens were measured for deflection over a 32-in. span. Most of these specimens were also measured for length change, weight loss and resonant fre- 394 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE February 1946 quency. Some of the free warping specimens made in the early part of the work were not equipped for length-change measurements. Cross sections of free warping speci- mens are shown in Fig. 34. See Fig. 35. Restrained warping specimens companion to the free warping specimens were loaded, as shown by Fig. 36, so as to prevent warping between the loaded quarter-points. Read- ings of the load required were taken periodically and, after the maximum restraint had developed, the load necessary to produce failure was determined. Fig. 32a (left) Comparator Fig. 32b (above) ^Restraining Bar Measurements of shrinkage-stress are made whether cracking occurs or not. The comparator (in Fig. 32a) is used to measure the changes in length of the steel restrain- ing bar that result from the strains placed on it by the concrete. From these measure- ments Hid the known properties of steel, the average stress in the concrete is computed Important details of the restrained-shrinkage type of specimens are shown in Fig. 32b (See also Fig. 23 in the tea thai the o an grip the bar only in the end- region; contact intl 2 tion is prevented by a thick layer of rubber. SHRINKAGE STRESSES IN CONCRETE 395 Fig. 33a Fig. 33b This method of test has been used to measure shrinkage-stress in specimens as wide as 12 in. and containing as many as four restraining bars. Those specimens that do not fail under shrinkage-stress alone are given additional stress with the machines shown in Fig. 33a and 33b. The machine at the left (Fig. 33a) is used for most of the specimens; that at the right Fig. 33b is used if the capacity of the other one is exceeded and if the specimen contains more than one bar. The load is applied to the bar, and the extension of the bar at the time the concrete fails is determined by the strain-gage shown in the pictures. The net amount of load on the concrete at failure is computed from the magnitude of the load and the strain-gage reading. The factor of safety is the ratio of the load on the concrete at failure to the maximum load represent od by the restraint against shrinkagi . "Sonic" testing: Most of the "unrestrained" specimens are tested periodically for frequency of vibration with the apparatus shown in Fig. 35a. Young's modulus is calculated from the resonant frequencies. ill 396 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE February 1946 sealed — -\ r r j" \-*--\ Fig. 34 — Cross section of free warping specimens. Fig. 35a Sonic testing Fig. 35b— Warping measurement SHRINKAGE STRESSES IN CONCRETE 397 Fig. 36a (above) Fig. 36b (left) Restrained warping 398 JOURNAL OF THE AMERICAN CONCRETE INSTITUTE February 1946 Warping, length-change, and moisture-content: On specimens like that shown in Fig. 35a, the change in length is measured with the comparator shown in Fig. 32 and the warping is measured with the curvature-gage shown resting on a specimen in Fig. 35b. Results are correlated with concomitant changes in moisture-content. Warping due to swelling: Previously dried specimens are placed, uncoatod side down, in a trough of water as shown Fig. 35b or ( bey are exposed to saturated air. The result- ing warping due to absorption at one surface is more rapid than that due to shrinkage and can ^ive rise to larger stresses. The specimens shown Fig. 36 are supported only at the ends. They are coated on all hut the bottom side and therefore as t bey dry they tend to bow upward. This tendency i> opposed b) th< shackles ai tin quarter-points which are connected to a lever system below, one for each specimen. The levers are held by the fine-thread screw-adjustment seen best in Fig. 36b. The screws are turned dow award until the force is just sufficient to prevent warping as indicated by the curvature-gage shown in both pictures. This i be moved from specimen to specimen. The force on the lever is measured periodically by finding the weight (bucket of shot) that will just hold the specimen in the position maintained by the screw. The force required reaches a maximum and iln-n n cedes slowly to zero as drying continues. When the maximum is reached, i he sp< cimens are loaded to failure. The ratio of the maximum >:nn\ to pn ping to that required for failure Lb the factor of safety. the factor of safety against cracking of Blabs of twice of the spe< - i quail} from both sides.